In today's lecture, I botched a derivation in which I was trying to
justify the relativistic momentum expression. This is basically just
the example on p. 1059 of Hecht (in Section 26.9), so you can refer to
that if you like.

Also, regarding the mass-to-energy transformation question as applied
to the relativistic KE, since KE depends on mass, an increase in mass
will correspond to an increase in KE (and also an increase in rest
energy, and therefore an increase in the total relativistic energy,
which is the sum of KE and the rest energy).