o you need a permit to climb some of the smaller mountains there? Is there anything that you can just show up and do with a partner without having to deal with local politics?
Yes, but none over 6000 meters.
My son and I actually climbed 13 smaller points or mountains while there last December that ranged up to 5600 meters. For peaks much higher than that you will need a permit. Chhukhung (not to be confused with Chhukhung Ri) is the highest point you can trek to without a permit. It is 5930 meters (19,126') high.
Chhukung Ri itself is sometimes listed as a trekking peak, but apparently only in error. Kyazo Ri 6186 meters is the actually peak that was supposed to be listed, but apparently someone goofed on the website (or so we were told). According to the National Park, you don't need a special permit for Chhukung Ri, in fact from their office and for a few rupees you can purchase a certificate for climbing it.
Lakhpa Ri, Island Peak?
You will need a permit for those ones.