I agree
Trying to match or create a similar buildup after crown was fabricated will simply hunt you down in the next couple of years.
Be honest you didn't do anything wrong and it happens. If I were you I would re do the buildup at a discounted fee, if tooth is endo treated I would definitely place a post.
Just explain to the pt the situation, he won't be happy but he'll appreciate it and rather than coming back after 2 yrs with an unrestorable tooth and blames you
Let us know how it went. Good luck
Look at your prep, add internal retention with a 35 cone bur or a 2 round bur. If your tooth is mostly core, you may want to consider endodontic retention but I wouldn't say "definitely place a post"... Posts in posterior teeth, if not done properly, can lead to problems.
Endo teeth are treated very well with removing ~2mm gutta percha from all the canals and to place composite plugs into the canals. In fact, amalgam cores that have internal retention and can engage the canals similar to the above are outstanding foundational restorations (along with RPD abutments, these are the only times I EVER place amalgam).
Remember, you are using a Bis-acryl material that has similar chemistry to composite resins and has a high potential to engage a resin core build-up. Furthermore, because you know it's a bis-acryl, you are probably not pumping the restoration (because you really don't have to) and therefore, if you have minimal taper to your restoration, it can become quite tight on the tooth, thus pulling the core. Use plenty of vaseline prior to using these provisional materials on resin cores.
BTW: re-do the impression, and don't charge for a re-do build-up... let the patient know what you did and how you want to make the crown fit great. Patient's get a little annoyed at having to get numb again, impression again, etc, but ultimately have more confidence in you that you are a good, caring dentist.