Is it a high side dimmer or a low side dimmer? In other words, do the instructions for the dimmer have the common connection between the LEDs and the dimmer on the ground side (which would mean it's a high side dimmer, as it is controlling the higher voltage lead) or the +12V side (a low side dimmer)?
If you have a low side dimmer, connecting the dimmer's +12V supply to the thermostat would probably not cause any problems because only minimal power is consumed there, and hence minimal noise produced; the 12V supply is only for the circuitry in the dimmer itself, not for the LEDs. However, in that case, you must have access to the ground lead from the LEDs for the dimmer to be able to control them; it won't work to switch/dim the positive line to them.
If it's a high side dimmer, I would avoid connecting it to the thermostat's 12V power, both from a noise point of view and from a current consumption point of view. From your description, it sounds like it might be a high side dimmer.
In either case, though, it really sounds like you'll probably need to pull a wire from somewhere (else).