You’re doing great: you let things unfold, you didn’t rush sex, he spent real time with you and your daughter, and he handled those introductions gently all very good signs. Sleeping over multiple nights and enjoying low-key activities (walks, movies, meeting your kid casually) shows comfort and real interest more than a flashy overnight trip would at this stage.
For his birthday: skip the overnight idea it’s too intense this early. Do something thoughtful and low-pressure that still feels special: a picnic at the beach (half-day), a homemade dinner + a small meaningful gift (a book, a nice watch strap, or a handpicked bottle if he drinks), or a sunset walk and a homemade dessert. Keep it in the $50–$100 neighborhood if you can meaningful, not extravagant and make the invitation about celebrating him, not fast-forwarding the relationship.
About commitment: you don’t need a label today. Trust the three-month rule: use the next few weeks to see consistency (does he follow through, include you in simple plans, show up for your daughter in small ways). A committed relationship looks like mutual initiation, consistent availability, and shared planning for the near future. If those things show up reliably, then have a calm conversation about exclusivity after a few months.
Want me to draft three birthday invite texts one playful, one confident, one casual that match your vibe? Also does your daughter’s dad have custody that weekend (so you can actually do a date-night plan), or would it be a daytime thing?