The immune system (and vaccines) operate mostly on the external bits of a viral particle. These bits perform several critical functions: hiding from the immune

The immune system (and vaccines) operate mostly on the external bits of a viral particle. These bits perform several critical functions: hiding from the immune system, attaching to cells the virus needs to replicate in, etc. Changing any of these things results in two questions. (1) Does the change make the virus better or worse at X? (where X is something related to what that bit was doing) (2) Does the change make the virus more or less visible / resistant to the immune system? The answer to both of these are independent, and for any one particular change, depends all the way down to chemistry and physics. It's possible that a given virus design cannot mutate external features without rendering itself ineffective. It's also possible that relatively minor mutations can make it more effective and/or able to avoid antibodies and the immune system. The mutation space has been studied for SARS-CoV-2, but to put it simply, it's a very complicated problem that involves (among other things) protein folding and molecular simulation. Not easy stuff to brute force.