Works only if you have Strike wallet?
Yes, unfortunately there is no other way. Someone could theoretically "share" their Strike account as an omnibus, but that would probably fall under the definition of a "money transmitter," and you'd need a license to do that.
But you guys are money transmitters, why not just forward the funds along to a third party wallet?
To get a money transmitter license, you must follow those pesky regulations - KYC, AML, etc. But maybe I didn't understand your original question. What would be the benefit of having this setup? You can connect BTCPayServer directly to a third-party wallet (and use Breez for LN liquidity). Strike provides FIAT settlements, so merchants are hedged against BTC price volatility and can easily withdraw money to their bank accounts.