My question is, without running new #4 feeders to the sub-panel, can I upgrade to a double-pole 70 according to 310.16 using the 90┬║ column?
No you can not use the 90 C column.
The reason is that none of your terminations are rated 90 C. The terminations will be 75 C at most.
However you could install a 2 pole 70 per 240.4(B) although the load on the conductors would have to remain under 65 amps.
I would install a new 70 amp breaker and look at balancing the load a little better.
With 53 and 43 amp readings you could try to balance it to about 48 amps each.
One other factor to consider is continuous loading.
With a 60 amp breaker the continuous load should not exceed 48 amps.
If you swap to a 70 amp breaker the continuous load could be 56 amps.