The main improvement I know of is in the room starting at 1:04, there is a faster route that is way more precise:    • Barbuta cool moves  

Aside from that, the problem is that the platforms triggered by the necklace seem to be on global cycles rather than per-room cycles. Notice at 3:42 that I get a bad cycle and have to wait around. In the section between the necklace and unlocking the key door, you're kind of at the mercy of the platform cycles. I was able to get more optimized runs of this section, but I wasn't able to optimize it enough to get a better platform cycle, so my runs kept ending up with really similar times.

I also got pretty good final boss RNG in this run. The number of times the final boss attacks before dropping down seems to be random. I think lowest I've seen is 4, and I got 5, 5, 4.

I got a 4m57s deathless run using basically the same strategies, minus the deathwarp. That run is here:    • Barbuta (UFO 50) Deathless Speedrun i...  

I have a proof-of-concept run, using autofire, showing that it is possible to get a faster platform cycle and that it is possible to 2-cycle the final boss:    • Autofire-Assisted Barbuta (UFO 50) Sp...  
But I haven't pulled off either one without autofire (and don't really care to try).