Hi, Hive account@rycharde, thank you for the problem and response. I should have triple checked before answering, especially when it was done by tapping solution on a phone and right before bed :p

The path 1-3 is flawed:
The last bit should be
2!+3!+3!=14, not a power of 2
2!+3!+4!=32 (2^5)
2!+3!+5!=128 (2^7)

Now the solution should be completed.

Thank you again for the interesting problem, I really enjoyed my time solving it ^_