Struggling with bottom bracket installation
Threadless cartridge BBs have their place, and that place is where the threads are destroyed. Thankfully you're not there yet. They are a compromise because they tend to have creak issues sooner or later no matter what.

You should take the frame to a shop with a proper piloted BB tap set. One side of your shell looks fine. The fix is to install the pilot into that side all the way and use it to start the other side. At that point one can use some force to confidently get through the cross-threaded section. (One shouldn't necessarily go all the way through with the second tap without partially removing the first, because referencing one side to the other isn't really how the tools are intended to work, but that's a tangent.)

You have enough good thread in there remaining that things should be fine at that point. The risk is continuing to mess with it and starting to lose threads completely, which could happen from the spot you're in now.

In the scheme of things, bike bottom brackets use a fairly long section of very fine threads. Its pretty common for low end cartridges such as Sunrace to not get things right and have problems with starting rough, going in rough, etc. It may work fine after chasing the threads, but if it doesn't, there's a lot to be said for just getting a Shimano UN-55. It will go in right and last a ton longer, and they've gotten really cheap.