electromagnetism - Does a moving charge have additional (magnetic) energy

My understanding so far:

A moving charge produces a magnetic field, B, in an analogous way to a current produces one.

A magnetic field has an energy density which is proportional to B squared.

My question is: Does a charged particle in motion have an additional energy associated with it's motion due to this magnetic field?

I suspect the answer is no, for a number of reasons, but want to check.

Additional work would then have to be done to accelerate a charged particle. This would be work against a force, and I can see no mechanism which could produce that force (other than just 'the increase in field energy').

If the charge is a point charge then the energy density would diverge close to that charge. (The volume integral would also diverge). So moving a charge would require infinite energy. I'm guessing this is similar to why you can't define the electric field energy of a single point charge, it doesn't interact with anything as it dwindles into infinity.