I recently saw this question posed on Facebook and it really made me think. There's a lot to unpack in it. First, nothing in writing is necessary. That's the great thing about fiction--as creators we're able to choose from a myriad of options. The only necessity is that a choice must move a plot (or subplot) to resolution. The real question is whether a writer's choice is logical.
So is "Bikini Leia"* logical for the plot and the character?
First, let's break down what's happening in the first act of Return of the Jedi. The gang must rescue one of the main characters from the clutches of the vile gangster Jabba the Hutt. Masterminding this is Luke Skywalker; which foreshadows the later conflict with the Emperor who is himself manipulating events on a larger scale. And while it's not explicitly stated: Skywalker has foreseen what is about to happen. That everyone is exactly where they need to be is an example of showing not telling the audience that Luke's ability to use the Force has grown since we last saw him. Leia is near Jabba so she can slay the Hutt, R2-D2 to get Luke his lightsaber, Lando to drive the skiff, etc. This tells the audience that Luke has learned to use his ability to see a future in motion well enough to successfully take risks with the lives of his friends.
Secondly, an alternative would've been to have Leia in the role of hidden pawn revealed at the required moment. Which meant we either had to have Lando completely absent from the first act after seeing him and Chewie take off to rescue Han at the end of The Empire Strikes Back--which would've short changed him. Or have him in Boushh's armor attempting to rescue Han in the middle of the night. The problem here is two-fold: the scene with Leia advances her relationship with Han while inverting the damsel in distress trope, and Jabba would most likely have fed Lando to the rancor rather than keeping him around.
Which brings us back to the original question: is Bikini Leia logical?
The answer is a resounding yes. Each one of Luke's pawns (to include Luke himself) face dire stakes when caught. If his plans fail, C-3PO and R2-D2 will remain the gifts of Jabba. Han, Chewie, Luke, and Lando face death. Leia's fate (because of her gender, class, and rank) is different and worse than death.
Jabba is part of a species heavily leveraged in the slave trade and we see a Twi'lek slave on his leash when we first see him. While Luke is the son of Anakin Skywalker, Jabba isn't aware of how valuable he is to the Emperor and Darth Vader. Chewie, Han, and Lando are just minor scum that at the end of the day few care if they continue to draw breath. So they're imminently disposable with little value as slaves.
But Leia is the last Princess of Alderaan, a high-ranking leader of the Rebellion, and a beautiful woman. Not only does this make her valuable from a cash value perspective; it makes her valuable as a trophy. By capturing such a powerful woman and putting her on display on his throne, her power becomes Jabba's power. She becomes a trophy that enhances his prestige. It raises her stakes in a way that's sadly all too grounded in reality.
Which contrasts with the tragedy of Oola, the Twi'lek slave girl we first meet at the end of Jabba's chain. She's a nobody from a species widely seen as a "slave species" existing only to be exploited by numerous depraved species that view slavery as part of some twisted natural order. Unlike Leia, she is disposable as she's fed to the Rancor for displeasing her master.
Indeed this is borne out in the costumes they're forced to wear. Oola's is revealing, but it's cheap and probably a hand-me-down. On the other hand, Leia's garb is ornate and made of precious (looking?) metal. The cloth of her "skirt" appears expensive. It is an outfit fit for an enslaved princess.**
But where Leia differs most from Oola and other damsels in distress is she is not passive when her back is against the wall. She will take her power back because she has been beaten down to not only survive--but thrive. We see this in A New Hope when she grabs a blaster and blows a hole in a grate to actively participate in her rescue. In terms of Leia's character development this is her being maximum Leia Organa. She reclaims her power by throwing her chains around her master's fat neck and strangling him with them.
This became the inspiration for the current expanded universe title: "Huttslayer". While there are some who think this is a modern interpretation, it's been there from the beginning. Going back to the 1983 novelization, there is depth of character in these sequences. When she's first captured she remains in control of herself:
Leia, on the other hand, stood tall before the loathsome monarch. Her anger ran high ...
... She thought about killing him outright, then and there. But she held her ire in check, since the rest of these vermin might have killed her before she could escape with Han. Better odds were sure to come later. So she swallowed hard and, for the time being, put up with this slimepot as best she could. pg 25
Later on Jabba's sail barge her inner strength is on display when the snake skinned sleemo advises her to "appreciate him" she's reflecting on how this is not the worst thing that has (or even could) happen to her:
He pulled her very near and forced her to drink from his glass.
Leia opened her mouth and she closed her mind. It was disgusting, of course; but there were worse things, and in any case, this wouldn't last.
The worse things she knew well. Her standard of comparison was the night she'd been tortured by Darth Vader. She had almost broken. The Dark Lord never knew how close he'd come to extracting the information he wanted from her, the location of the Rebel base. ... pg 41
The fate here worse than being Jabba's "dancing" girl is that she almost broke and gave-up the location of the Rebellion. She values the lives of others over her own wellbeing.
And then finally we get to the end and the death of Jabba the Hutt:
She jumped onto Jabba's throne, grabbed the chain which enslaved her, and wrapped it around his bulbous throat. ...
Yet Leia's hold was not merely physical. She closed her eyes, closed out the pain in her hands, focused all of her life-force--and all it was able to channel--into squeezing the breath from the horrid creature. ...
... as Jabba wildly thrashed, frantically twisted from this least expected of foes. pgs 44-45
There is a lot going on here! She's a slave who is literally using her chains to set herself free--the imagery obvious even to an 8-yo. She's demonstrating that she is also strong in the Force. And finally we see that Jabba's speciesism and sexism led him to underestimate a small human woman and thus to his doom.
So while Lucas may have had other options, this choice fits with the story. It's logically consistent with the plot and character. That Jabba wants to posses her and display her as trophy is a testament to her strength. That his arrogance blinds him to the threat she poses only highlights how stupid it is to judge a book by its cover.
Citation: Kahn, James. Return of the Jedi. Book Club ed. New York, NY: Ballantine Books, 1983.
* I used the term "Bikini Leia" instead of the traditional "Slave Leia" to stay true to the original language of the question. My preference is to use "Slave Leia" over alternatives like "Bikini Leia" or "Huttslayer". She was enslaved and while she could have fought, and died, she choose not to and wait for the best time to break her chains. To attempt to soften the language deprives the sequence of its gravitas and ultimately robs the character of her agency.
** Whether intentional or not on the part of George Lucas, there is A LOT going here regarding not just sexism but class and race. Leia is royalty and human and on the sail barge we see him loosening her chains a little to let her wander. He speaks to her as if she has a choice to "appreciate" him. This runs counter to what we see with Oola who gets barked at and ordered to do things. When she displeases him; the floor is literally pulled out from under her.