Well that’s easy enough to do. I’ve made those before. They mostly work as particles, afaik, but I think it’d be fine.
On the other hand…
Rendered image before compositing
I think blurring solid spheres might work. But billboards would too. Just render out a white fuzzy circle either with or without alpha, but if you don’t do alpha make sure it’s on a black background. Use the alpha channel/black and white image as a mask. Then use the colors generated by the particle information to color it in.
To make billboards, it’s as easy as putting a copy rotations constraint on to the object you want billboarded, making it copy all axes, and setting the camera as the target. A locked track would work as well, possibly even more accurate to a sprite.
Anyway, I hope this helps you. Your prerequisites required some thinking, but I think it will work out.
If you’re ever able to, EEVEE and Cycles can produce some pretty amazing results, and I highly encourage you use them. But BI has some stuff in it that’s pretty unique, and I keep coming back to it every now and then.