Yes, a "dual-band" beacon for 22 meters and 11 meters would be awesome. I'm looking at the frequency list for currently active 22m beacons and most of the beacons seem to be in the 13554 kHz to 13558 kHz range with a few others higher in frequency (I know most ISM QRM centers at 13560 kHz), then 13562 kHz or so up to 13566 kHz/13567 kHz. 22 meters 13553 kHz - 13567 kHz (x2 is 27106 kHz - 27134 kHz) or roughly CB channels 12 (27.105 MHz), 13 (27.115 MHz), 14 (27.125 MHz) and 15 (27.135 MHz).
Looking at the second harmonic of those frequencies and their relation to the 11 meter/CB band plan:
13553.0 kHz x2 = 27106 kHz / 27.106 MHz - 1 kHz above CB channel 12
13554.0 kHz x2 = 27108 kHz / 27.108 MHz - 3 kHz above CB channel 12
13555.0 kHz x2 = 27110 kHz / 27.110 MHz - right between CB channels 12 and 13, and probably a better spot for an 11m beacon
13557.0 kHz x2 = 27114 kHz / 27.114 MHz - 1 kHz below CB channel 13
13560.0 kHz x2 = 27120 kHz / 27.120 MHz - right between CB channels 13 and 14, but suffers from 13.56 MHz ISM QRM)
13562.0 kHz x2 = 27124 kHz / 27.124 MHz - 1 kHz below CB channel 14
13562.5 kHz x2 = 27125 kHz / 27.125 MHz - right on frequency for CB channel 14
13563.5 kHz x2 = 27127 kHz / 27.127 MHz - 2 kHz above CB channel 14
13565.0 kHz x2 = 27130 kHz / 27.130 MHz - right between CB channels 14 and 15, another good spot for a beacon
13565.5 kHz x2 = 27131 kHz / 27.131 MHz - also another good spot for a beacon
13566.0 kHz x2 = 27132 kHz / 27.132 MHz - 3 kHz below CB channel 16, probably another good frequency
13567.0 kHz x2 = 27134 kHz / 27.134 MHz - 1 kHz below CB channel 16, maybe not the best choice
So the best bets for dual-band 22m/11m beacons are looking like
13555 kHz (second harmonic: 27110 kHz) and nearby freqs 13553 kHz - 13557 kHz (second harmonics 27106 kHz - 27114 kHz)
13560 kHz (second harmonic: 27120 kHz) and nearby freqs 13558 kHz - 13562 kHz (second harmonics 27116 kHz - 27124 kHz), although your 22 meter beacon wouldn't do so well with all the ISM QRM on 13.56 MHz
13565 kHz (second harmonic: 27130 kHz) and nearby freqs 13563 kHz - 13567 kHz (second harmonics 27126 kHz - 27134 kHz)
Your best bet on 11m (as far as a dual-use beacon goes anyway) is to stay as close to the halfway point between the CB channels, so 27.110 MHz, 27.120 MHz and 27.130 MHz. CB channel 14 on 27.125 MHz is doable and 1-2 kHz either way from the 27110/27120/27130 center frequencies probably wouldn't hurt too much either. Plus, 27125 is just easy to remember (22 meter frequency 13562.5 kHz).
I know there was a Part 15 beacon in operation on 27.120 MHz (maybe it was 27.125 MHz?) at one point or another.
I strongly support your plan to do multiple beacons. You could do 2 MHz, 4 MHz, 8 MHz, 13 MHz/22 meters and 27 MHz/11 meters.
If you decide to do a dedicated beacon for 11 meters then I would recommend using one of the four R/C frequencies (26.995 MHz, 27.045 MHz, 27.095 MHz, 27.145 MHz or 27.195 MHz) or going slightly above or below the CB band to get a clearer frequency. Of those frequencies, 26.995 MHz, 27.145 MHz and 27.195 MHz are probably your best bets. Since those frequencies are used for data transmission, a low powered beacon transmitter would fit right in. Staying within the CB band (even if you use a dual-band 22m/11m beacon transmitter) will probably reduce your chances of being noticed.
Under FCC rules, you can transmit data/telemetry on 26.995 MHz, 27.045 MHz, 27.095 MHz, 27.145 MHz, 27.195 MHz or 27.255 MHz. 26.995/27.045/27.095/27.145/27.195 are data-only channels and have a 4 watt maximum power limit under Part 95 of the FCC rules for the Radio Control Radio Service. 27.255 MHz is also CB channel 23, but...under FCC Part 95, the maximum power goes up to 25 watts for professional-level remote control, paging, datalink and telemetry systems operating on that frequency. There are several known systems using 27.255 MHz and paging and FSK datalink / telemetry signals have been heard on 27.255 during band openings, making those telemetry and other systems de facto 11m beacons.
Or, if you want to stay out of the QRM from within the CB band, use 27.500 MHz, there are other beacons operating on that frequency as well.