Using the Raduino, the LD78L09 is no longer needed for VFO stability, so you can likely go below your 10.7v
Will lose dynamic range in those bidi amps as you get too low, but I wouldn't be surprised if we still have RX at 6v, with considerably lower current draw. ?I plan to play with that as well. ?At 13.8v, Q13 will be getting awfully hot as is currently (ha!) being discussed here in a separate thread. ?A buck-boost could create more trouble on rx (noise) than it would solve. ?I think I saw where the new units going out now are around 70ma on rx with 12v, down from 90ma.
There are switched cap voltage doublers out there, but I'd go for a monolithic boost switcher (with inductor), turn it on only during transmit. ?LinearTech and TI both have many to choose from, LTSpice is free, and a great tool for modeling their switchers. ?In fact, it would likely do a good job of modelling the entire Bitx40.?
Alternately, have two batteries, in parallel for QRP and switched in series for QRO, as suggested here elsewhere.
Jerry, KE7ER