I think standard servo's will not give you enough precision.
if the servo has 1° precision and the rod is 20 cm long you get 0.349 cm precision. And don't get me started on servo's and repeatability.
You will have to go for a stepper with reduction to get some decent results.
I thing the heating issue caused by the long rod is an attention point but not a show breaker. I think your proposal is a good starting point.
Moving the bottle in a 3 axes CNC machine solves the long nossel precision problem but cause new problems. IMHO the biggest one will be to have the partially printed object properly fixed to the bottle.
Now I think about it. It is probably better to have the whole printing head and swivel joint in the bottle. The Z axes outside the bottle and use the neck of the bottle as a solid fixing point.
Best regards
Jantje