Does disposal order prevent me from playing Rhasa after from Last Rites after already paying it cost?

No. Once you've already paid Rhasa's cost, Disposal Order can't stop it — Rhasa left your trash the moment the play began (Rule 354, step 1: move to the Chain) and a finalized unit resolves immediately (Rule 337.2), so it's no longer a card "in your trash" for Disposal Order to recycle.

The window to stop it is before the Last Rites trigger resolves. Your opponent can play Disposal Order as a Reaction while the trigger is still on the Chain; if Rhasa is chosen, it gets recycled to the bottom of your deck and the trigger then finds no Rhasa in the trash to play (same logic as FAQ #11060 on Fizz). One more side note: if Disposal Order recycles 3 other cards before the trigger resolves, Rhasa's discount shrinks and it'll cost you more when you do play it — but if you've already paid, none of that applies.