"Could we have become more than friends?" I don't think so. She walked in knowing what she wanted from him and what she was willing to offer. She seemed used to selling her body and was unhappy with her home situation.
Glossing over what she really wanted, huh.
So he realised that, and yet he didn't try correcting her and just went with "Friends"? Man, you missed quite the opportunity.
He seems to realise that, too. "If I had chosen my words more carefully, could we have been something more"? Yes, yes, and YES. I think so, anyway.
By just going for it and accepting her request of 'friendship', it basically just made your relationship physical with the justification of paying to let her stay there with you.
A simple 'We can become lovers and then you can come over anytime/stay as long as you want' would have gone a long way, I wager.
Granted there are some key elements of dialogue throughout the story that just might reveal Fumina-san's real motive for *a-hem* imposing herself uopm Tsukamoto-kun as was depicted (no spoilers from me). As to where this current arrangement may go beyond these 24 pages, perhaps only a ... wait for it ... sequel might tell. Whether one shows up or not, we'd all do well to keep in mind the following bit of advice (some might recognize what follows):
FRIENDS ... how many of us have them?
FRIENDS ... ones we can depend on.
FRIENDS ... how many of us have them?
FRIENDS ... before we go any further, let's be FRIENDS.