Hmm. That sounds like a problem, because there are only 10 available "memories" available for handsets.
I don't have my schematic with me, but I seem to recall that these codes (or keys really) are stored in a non-volatile memory, which does not erase unless it is done specifically using the de-register process. That is why these things can be make and stocked for many months (without power) and still not have to register the original handsets that come with the base unit (whether it is 2, 3, or more).
I got my set from Costco, when they were getting rid of their final stock, and I am pretty sure that the unit was at least 9 months old, and I didn't have to register the handsets that came with them.
I kind of think that you may be "stuck" with what you have, or what you mentioned in your first message, which would be to get a refurbished set (or just the base). Just be careful that if you get the base by itself, that all of the slots have been cleared so that you can register all the handset that you have.
I actually have two base units, connected to two different phone lines, with 4 handsets registered to each base. But they do sometimes get "crosstalk" when both lines are in use (I can hear my wife's conversation on my handset, but it is faint). I actually talked to the uniden tech-support folks about this, and this is NOT supposed to work, but the two base units are on opposite ends of the house. So I think I am just "getting lucky" most of the time.
Sorry to be so long winded. I wish I could be of more help, but I think a second base unit (at least) would be your best choice.
Please let me know what you decide.
Take Care & Good Luck, Greg A.