You didn't mention how large the original file is or what bitrate it uses. If you want to keep it that size you'll have to alter the render bitrate to the same amount. I would think a 4:19 minute 1080p mp4 file at a decent bitrate would be exactly what you got, around 350mb. If you want to go smaller you'll sacrifice quality but I'm guessing the source file you're working with is already lower quality if it's significantly smaller.
Either way, the solution is if you want a smaller file, lower the bitrate of the render.