Welcome to Healthcare magic.
I am sorry to hear of your multiple issues.
To begin with there should not be any issues with the actual surgery being performed while having a short Achilles tendon.
However, since after the surgery, during the recovery phase which is roughly 6 weeks (getting back to normal tight and slim shoes may take months), one has to avoid activity and even when walking one has to do heel walking which would be a difficult task for you considering the short tendon.
And even if you find a way round that still the heel length which is usual for you, you may need to decrease it, once your foot has healed from the bunion
surgery, and since your Achilles tendon is short, this may cause an issue.
So what you can do is discuss with your doctor the procedure to correct the Achilles length (ATL=Achilles tendon lengthening) before going for the bunion surgery. And besides that the short Achilles itself can be a cause for the bunion formation.
Hope I could be of assistance
Wishing for you the best outcome