Help with a few snags...

So I’ve run into a few issues as I’m trying to get ready to assemble a bed I’ve been working on. Basically I’m down to cutting out a groove groove for the 1/2” plywood panels. My “workshop” is in my apartment, so a router and other power tools aren’t really an option. I got a plough plane to cut the groove, but was disapointed to find out that the groove it cuts is a bit bigger than the plywood.

So my first question is this. How much space should there be in that groove? Do I want it nice and snug, or do I have to leave any room for plywood to expand – thickness-wise?

Second, I think I’ve got way too much room, and I’ve heard of people using tacks to hold a panel in place… I’d rather not, but is there any other good option?

Finally, I have a smaller blade, but it’s too small, but I could cut a groove then enlarge it I suppose. Does anyone have any suggestions on a good way to enlarge it? I suppose I could do it with a chisel, just not sure how pretty that would end up, or if I’d ever finish. Or, I also think they make some kind of specialized plane for that, but I’d rather not invest in another one if I can avoid it. Another option, I could try to cut two half sized grooves overlapping to try to get the right size, but that might be a bit tricky

What would you do?
Thanks everybody!