If I am about to deploy with a very unit short on NCOs, can I waive BLC until I redeploy?

No under STEP you must attend BLC to be eligible for promotion to SGT. If you want to get promoted then you must attend BLC. I would try and go before you go. There was a time when they would wave that I went to WLC with a SSG. They may change the rules but right now you must go to BLC to get promoted even in a star MOS.

BLC is taught in Kuwait.

Look around stateside and see if if you can walk in. Many people walk in at Camp Shelby. It's an OASS school and active come there from 7th group and other units all the time.

As others have said, getting BLC is one of those steps of being an E5 that is more of a requirement these days. While SPCs are the vast majority of E4s in the Army, it seems that the rank of CPL occasionally is forgotten. The Marines use them quite effectively and they are just as much an NCO as a SGT. My last active duty unit was predominantly infantry, and it took me a bit to see junior enlisted go to parade rest for a CPL since the first 4 years of my Army career I didn't have them around me. You can always see if lateral promotion is an option, which will help prepare you for being a SGT.