The patient is a shower stall, acrylic base, cement backer board, 8x10
tiles in thinset. The tiles overlap the lip of the acrylic base by
1/2"(?) and there is a 1/4" gap between the bottom row of tiles and
the flat top of the base. I went to start grouting, and noticed that
the gap behind the bottom of the tiles isn't always filled with
thinset.

Is it necessary/desirable to try to push thinset/grout under the
bottom tiles and into the gap behind? Should I just squeeze extra
caulk up into the gap? Caulk seems easier, but won't support the
bottom bit of the tile as well.

Any advice appreciated.

Red