I'll have to go look again to check my memory, but I thought I remembered it not costing to transfer FLR to a new passport, but if you do it with ILR it costs a ton and makes it worth carrying two passports. If that is the case, I would get it transferred beforehand and save yourself the grief.
Okay, I just looked. The info is one of the earliest posts in the visa and citizenship board of the forum--one on transferring FLR or ILR to a new passport. You will get better information there. If I were you, I'd look strongly at changing it now while it's free as an FLR, but you will have to do it in person at a Public Enquiry Office. You can't do it by post anymore.
Excuse me if I got this mixed up. I'm still sorting this out myself. I was going to just expedite my passport that is up for renewal next year, but the idea of being stuck waiting for my US passport to come back while DF has taken time off work so he doesn't have to leave me is beginning to make me nervous. I think I may just break down and go this route as well then either transfer the visa or carry two passports.