No stop making excuses for this. I hate it when people make these mealy-mouthed justifications for this kind of cuckoldry. He can at least make some plans moving forward. Or, and here's a crazy idea, he can get his newfound girlfriend involved in information-gathering since she has the perfect cover.
At this point, the plan should be:
1. gain more strength,
2. gain more authority based on that strength.
Without more authority, it looks like they can't even leave their quarters, other than for training.
Sending her to do anything is way too risky. There is no guarantee there are no hidden cameras or other forms of sensors (because MC doesn't know anything about technology), or people with stealth skills watching the maids 24/7 (no knowledge about world power/military power of the country). Or she can just get asked whether she was doing "anything a maid is not supposed to do", and forced by the slave brand will have to confess everything. Any plans must absolutely be hidden from her until she gains freedom.
I get that none of those points are actually made by the author since it's stated that MC basically only cares about seggs atm, but coincidentally he is doing the best thing he can - getting stronger, and getting FMC stronger in secret.