Am I right to say that this panel should be fed from 350's because of the temp rating on the lugs?
So if my above suspicions are correct, can I fix it like this: There is a J-box relatively close to the panel. Can I just remove the 300's back to the J-box and split bolt 350's from the panel to the j-box? I guess that is assuming the lugs on the other side are 90C rated, which come to think about is unlikely.
You may be alright the way it is depending on the load. If the calculated load is less than 570 amperes, I do not see a problem with using 240.4(B). By the way, you will not be able to use the 90°C rating of the conductor except for the purpose of derating. I do not believe there is any way you will be able to terminate at anything other than 75°C at that ampacity.
The temperature rating of the lugs, themselves, is not important. It is the temperature rating of the terminations that is important. The termination includes the lug, the device, and the enclosure. All of these factors are important.
I once had a contractor argue that because the lug supplied in a Square D fusible switch was rated AL90CU he could use the 90C column. But he was wrong. There is no UL Listed (or any other NRTL that I know of) overcurrent protective device with a termination temperature rating in excess of 75C.