Lots of people have been in your situation, but I will say this - get British citizenship. You never know what may change for you, and as you know, if you are out of the UK for more than 2 years, the ILR expires. It doesn't affect you US citizenship, and it makes everything a LOT easier when you travel to/from the US. It's money well spent, and eliminates all sorts of potential hassle at the airport. You use your US PP to enter and leave the US (as required by law), and your British PP to enter the UK.
In the interim, renew your passport as usual, and carry both your new and expired PP with the ILR sticker. Someone else may pop on and include info about the BRP that folks now need to have. My ILR predated that, as does yours. Not sure if you need a BRP or not.