I would think a full-size bed would be fine, with bed rails. My oldest was 19 m/o when my younger child was born, and I found a toddler bed at a yard sale that we used for a while. I got it for $20, with the mattress, so it wasn't a significant amount of money. When I moved my little one out of the crib (he was climbing in and out, so I just said bye-bye to the crib), I just put both boys on the crib/toddler bed mattresses on the floor, and they're still there -- aged 4 & 2. When they get too big, or if I get pregnant or something else changes, I'll probably put both of them into a full-sized bed.
My brother's boys have shared in a full-sized bed for years (they're 7 & 5), and I'm pretty sure my brother made the same transition you're contemplating -- moving the older one who is still fairly young (his boys are also 19 months apart in age) into a big bed so the little one can have the crib.
I wouldn't get a toddler bed, unless you can find one used for cheap. You may decide in the future to just put both boys in one full-size bed. It seems to me that you can bring out a "new bed!!!" for your older boy and he'll be happy, even if it is smaller than his other one, unless he doesn't like the idea of matching his younger brother. I'd play it by ear, and not spend too much money.