As you've said, attempting to do a "visa run" to another country will almost certainly not work. There are specific rules for Canada (and Mexico and some other nearby countries), but even if he was to go to another country then he would most likely be refused entry on returning to the US.
The ONLY option for staying in the US for more than 90 days is to apply for a visa.
However given your circumstances, it's very possible that a visa would NOT be approved. The US frequently refuses visa applications from people from Visa Waiver Program countries, unless they have a specific need that requires them to stay in the US for more than 90 days. The fact you are a US citizen doesn't necessarily help his case here, as it could be seen as you moving back to the US and him living there with you.
If he does apply for a visa and it is refused then you unfortunately may end up in a far worse spot than you are in now, as he will need to declare this visa rejection when applying for an ESTA to use the Visa Waiver Program, which may lead to his ESTA not being approved meaning he can not travel to the US under the VWP (See https://help.cbp.gov/app/answers/detail/a_id/1097/~/previously-denied-a-visa-or-immigration-benefit)
Your best option might be to reach out to an immigration lawyer who will be able to provide the best option - be that risking applying for a visa, staying only for <90 days, or potentially getting married at which point he would be able to apply for a visa as your spouse.