Thanks for posting all the images of your item , sapphire (and welcome to the forum) .
As some others have mentioned , perhaps a professional cleaning should come first (before framing) ... sometimes even the best treatment of items will cause a bit of further damage (to fibers , dyes , etc.) ; time-to-time , repairs need to be done post-cleaning (and pre-cleaning).
Doing so would avoid the future problems of dismounting , disassembly & re-doing the whole framing/blocking process .
I just think it's easier to do it right the first time ... also think it's worth it to 'take your time' doing so .
Once you're done , you'll have something that you won't have to hassle with & can sit back & enjoy !
I will also add that there are other , somewhat inventive & time-consuming methods of framing/matting fabric items , besides shadow-boxes .
Let's face it , a frame is intended to 'frame' an item within it ... I say choose the type/style of frame that you'd prefer your embroidered be shown in , for it's lifetime .
I think a frame can certainly add to , or subtract from the appearance of a finished unit .
Considering the facts you've related about the fragility of your item , I'd agree with the other members who posted about mounting yours on another piece of fabric , professional cleaning , acid-free mat'ls , and of course UV protection .
Wishing you all the best with your project !