I missed out on one because some careful eyes caught my birthday on one of my transcripts, and it wasn’t me :( I didn’t know they put my birthday on there and I never noticed it before lol.
It couldn’t hurt to ask, but I would consider the opportunity as a no go and look for other paths or other opportunities. Even though you may have been rejected from SULI you may have noticed that these national labs have lots of internship opportunities even directly through the national lab too (not just through SULI program) so you could probably look on their website and see if you can score something through that; maybe find a different education program to sponsor your time there. The other thing you can do is reach out directly to the PI's you knew and see if they have different means or funding (but don’t hassle them).
You can also look for internships elsewhere and diversify your experience. The year I got rejected I ended up getting another opportunity at NASA, which was awesome and worked out really well. The things I learned there later became critical for my current full-time career. Don’t give up- there’s something out there and you can do it :)