They stay hot after shutdown and you'll still need to cool them to avoid damage. If you use your propellant/coolant for that, the ISP will drop further.
You plan for that in your burn, essentially doing a very low-powered long tail in it. Yes, doing this will slightly reduce Isp, but assuming you can alter the size of the throat on your bell, this effect is just single-digit seconds on the entire burn, compared to the "hundreds of tons of radiators" the GP referred to.
A variable geometry nozzle would help a lot here, by keeping a lower volume of propellant to still be ejected at a high speed. May be a good idea after all.
They stay hot after shutdown and you'll still need to cool them to avoid damage. If you use your propellant/coolant for that, the ISP will drop further.