I had the same problem with placing DS7 for homeschooling last year. Don't panic!
Some of it is just an adjustment in how we think about teaching. I gave DS7 (then 6) the Singapore placement test, and realized he was teaching himself the math--math he'd never before attempted to do!--from the test questions! "How am I supposed to work with THAT?!?" I thought.
But you figure it out. It might be good to have a conversation with your DS about how Mom is learning right along with him, and so if HSing is going to work, you're going to have to work as a team. "I'm bored," isn't a very helpful, team-oriented comment. More specific info about why he's bored might be helpful though.
Consider that it may be the subject matter or form of teaching you're using more than the level of work that's boring him. It might be useful to take your DS to the library and choose what he wants to read or do. For math, maybe you buy a bunch of Singapore workbooks at levels ranging from 2-5, say, and just let him do whatever he feels like doing. Or maybe you need to let your math work grow out of science experiments or something. Not all kids like workbooks. Not all kids need to follow a curriculum in a lockstep form. (Said the control freak who is still learning to let go...)
The first year is a time of experimentation. You WILL make mistakes about placement and style of teaching. It's natural and virtually unavoidable. If he understands that you're learning, too, and that trying your best for him, the mistakes won't matter.
Finally, I strongly recommend that you don't worry about gaps. He's 5! Math at the K-3 level is pretty doggone basic arithmetic. There's not a lot of room there for gaps!
Your goal for this year should be to challenge your child, to do your best to make learning fun. Not "party-time" fun, but "I have to work at this" fun. If that means that you give him the level 4 workbook (or 3 or whatever), then so be it. You can always check out the lower level workbooks and slip in bits of them if you see that he's missing something along the way. I doubt if you'll need to do much with them, though. If a kid can add 4 digits, he doesn't really need to practice adding 3 digits. Even in Singapore--which doesn't have much repetition--a lot of the curriculum is this sort of move up from step 1 to the slightly-longer-but-same-idea step 2.
As for testing, if you can get the school to do it, then you absolutely should. Achievement testing has been useful to me. But I wouldn't bang my head against that wall for too long if it isn't moving. If money is an issue for the school, then it may just be a lost cause. You can figure this out without it if you and your son communicate well.
All the best to you! HTH!
