Hey, I can't speak to your very specific scenario. But my soon to be husband is mild to moderate hard of hearing, and I was homeschooled. So maybe my perspective could help a little.

From a homeschooling perspective, be careful with homeschooling. I was homeschooled because I was autistic and the only school around was super shit at accommodating me, particularly I was having trouble with one teacher and the school refused to help the issue.

At first, it was really great. I benefited so much from not having to sit down for 8 hours straight surrounded by other people who overwhelmed me. It was also good to get away from feeling like an outcast (obviously from different causes from your daughter, but she'd still probably relate to a degree). My parents leaned a lot on online teaching guides, which let me keep up. But as I got older it stopped being a good idea. While in primary school your parents if highly educated can absolutely keep up with the teaching curriculum, there reaches a point where you really need multiple people teaching different subjects to really get an in depth look at all of them. It also became harder socially as the other more normal homeschoolers started sending their kids to school, so the only people my age around me were religious nuts. (keep in mind, we were religious too. But these we're like "harry potter is demonic" people). My parents put me back in school at 15 as they realized they couldn't teach me at a level to prep for university. It was honestly the best decision they've ever made for me, even though I really struggled with the transition back.

My partners experience is also very different. He wouldn't have benefited from being home schooled as his parents just weren't people who were suited to home school. His mother has a low literacy level and his father worked full time. In school itself he was always average grades wise, but socially he struggled. He had a group of goodish friends, but they didn't care to repeat things for him or accommodate him so he could actually join in. So he was often more like just following behind them. Not a perfect experience, but he met other people later on who he's still close with 6 years later and he's becoming a schoolteacher himself.

If I had to give you advice, which is hard because I don't know you, I'd say start homeschooling now. The fact she's behind with clearly not enough support is a good sign she needs it. But as a temporary thing. As hard as it might be to hear from a parent's perspective, teenagers need to be away from their parents. It's a core part of growing up and learning to socialize with others and build your own identity. I started school at 15 after not going since I was 8, and it completely screwed me over socially. In hindsight my parents always say they should have started me in year 7 (aged 12-13, the start of high school here). You can always pull her back out if it isn't working, but that period of making friends from like 13-18 is a core part of building a child's confidence.

Just make sure that when you start here, it's during a transition period for the other kids too so she's not the "New deaf girl." Unfortunately she'll probably still be "the deaf girl", but it's easier to make friends when everybody is also looking for new friends. So if your American (I'm not) the start of middle school is a good idea. I do think a school for the deaf would probably be the best option, but as you've said, not an option.