I've done a little more research on this.
There are frequencies restrictions but nothing other than what the "regular" rules say as you mentioned. A cross band repeater is actually a repeater under the rules and so has the same frequency restrictions as a repeater:
97.3(a)(40) Repeater. An amateur station that simultaneously retransmits the transmission of another amateur station on a different channel or
channels.
97.207(b) A repeater may receive and retransmit only on the 10 m and shorter wavelength frequency bands except the 28.0–29.5 MHz, 50.0–
51.0 MHz, 144.0–144.5 MHz, 145.5–146.0 MHz, 222.00–222.15 MHz, 431.0–433.0 Mhz, and 435.0–438.0 Mhz segments.
This repeater station requires a control operator who in this case is the person with the HT. Also, the person that is controlling the cross band repeater from a HT is doing so remotely and is an auxiliary station and as such is restricted on what frequencies they may use to control the remote station (cross band repeater):
97.3(a)(7) Auxiliary station. An amateur station, other than in a message forwarding system, that is transmitting communications point-to-point
within a system of cooperating amateur stations.
97.201(b) An auxiliary station may transmit only on the 2 m and shorter wavelength bands, except the 144.0–144.5 MHz, 145.8–146.0 MHz,
219–220 MHz, 222.00–222.15 MHz, 431–433 MHz, and 435–438 MHz segments.
The auxiliary station definition and rules are vague, but that is what the ARRL calls the person with the HT using a mobile as a cross band repeater:
http://www.arrl.org/auxiliary-station-faqI think I figured out a potential but cumbersome solution to the ID problem. The original transmission from my HT through the cross band repeater covers the ID requirement for part of the repeater operation. To cover the other side I could switch frequencies to the band that I was using as an output from the repeater and ID back through from the repeater output back to the frequency link going to the HT. As long as I do that every ten minutes I'm fine.