Honestly, this is one of those Meta Business Suite gremlins that drives me up the wall. I've run into this exact situation at least three times in the last quarter alone, and each time it's the same bloody loop - you've ticked every permission box, the partner has accepted, yet the ad account refuses to show up in their selection dropdown.
Here's the process I've landed on after a lot of screaming at the screen:
Double-check the role assignment.
Go to Business Settings > Accounts > Ad Accounts. Click on the account in question, then the Partners tab. Ensure the partner's business ID has Manage Ad Account or Ad Account Admin level - not just standard access. The dropdown gremlins seem to ignore anything below Admin.
Resend the invite from scratch.
Remove the partner entirely from that ad account's shared list. Then re-add them, making sure you toggle "Manage Ad Account" on. Have them accept again. Sometimes a fresh invite flushes the cache.
Check the business portfolio hierarchy.
If your partner's Business Manager is owned by a different root business, the ad account might not propagate into their 'Shared with you' list. I've had to get a rep involved to manually sync it on the backend once. Painful, but works.
Test with a secondary user.
Try adding a colleague from your own BM as a test. If they can see it, the issue is on the partner's side - usually a permissions conflict in their own BM settings.
If none of that works, it's likely a Meta platform bug. I've had success opening a chat with support (yes, the dreaded chatbot) and requesting a manual refresh of the ad account's visibility for that partner. Took two days, but it sorted itself out.
Keep us posted if you crack it - always keen to hear what actually does the trick