I agree with @wreklus that overstimulation and general social anxiety can be an issue. I think you need to focus less on your desire for him to get out more, and more on how getting out might be rewarding for him. And also keep an eye to structuring excursions that are comfortable and safe for him.
What did he enjoy before he got ill? How does he spend his time now? How can his interests past and present be enhanced by getting out and interacting with the world and the people in it? What’s the payoff for him? You are asking him to risk something, but he’s got to see a reward, or it will be much safer to stay inside.
In my case, I wanted to learn to sing. It grew out of some delusions of grandeur while psychotic. So I audited a group voice class at a local college and joined a local pops choir. Initially these were stressful and scary for me, I often returned home after sessions and sat up all night overthinking or crying about my perceptions of events. But I kept coming back. Sure I sometimes skipped practices or classes because I didn’t feel up to it. But I was ego-invested in a payoff, and I’d paid for the class or the music or whatever. Looking back on it, it seems a bit silly to me now, but at the time it was a brave thing to do. But I’d gotten to the point that I realized that my life wouldn’t improve if I didn’t put in some effort.
It’s not that much different than what “normal” people do to “put themselves out there”. You just have a few more obstacles. But one thing that your boyfriend has that I didn’t is someone to go out with who he can trust. I slogged it out alone for years until I made enough friends and found people I trusted. You should leverage this as much as you can. It’s particularly difficult to navigate social situations when you can’t trust your perceptions, or are preoccupied and have trouble keeping up with what’s going on. Having another set of eyes and ears that can reality-check for you is gold in these situations. I found it much easier when I found a girlfriend who could help me with such things and serve as a social buffer. A supportive partner is helpful to anyone, but doubly so to people with MI. So I’d look to his interests and goals and see if you can find a way to pursue them together somehow. Taking baby steps if you can.