Is there any issue with these wires being that close to insulation?
I would intend to run the new wires like the old ones (between the
two layers of insulation).

You aren't supposed to insulate around the old cloth covered "knob and
tube" wiring, but there are no problems insulating around modern romex
cables. Wires run through insulated walls all the time.

As for the temperature ratings on NM type cables, that mostly pertains to
the heat buildup around light fixtures. The newer NM-B cable has a higher
temperature rating than the regular old NM. I've opened up old light
fixtures that used regular NM cable, and the insulation was dry, brittle,
and cracking away leaving exposed wires.

In any case, buy new cables today and you'll get the higher temperature
rated NM-B (doesn't hurt to check the printing on the cable though). Either
type will be fine in insulation though.

Anthony