I'm not sure of the current model numbers, but I have had two Epson printers purchased for less $100 that did a great job on 4x6, 5x7, and even 8x10 or ltr size. I am going to make an outrageous assumption here, but people buying 4x6 and 5x7s are looking for and buying memories, not fine art. Take a look at a couple of them, they're fairly quick at the smaller sizes and even have a reasonable quantity of ink. You can probably make a bit of change with these.
I'm sure Canon and HP also have some workhorse smaller printers that can handle this job. Thinking back on it as I type this, I've sold smaller prints up to 8x10 off these low end printers and I hardly sell anything. I have a Canon i850 on my desk right now that is my day to day document and occasional photo printer and I can't even remember when I bought it.
My regular photo printer is an Epson R1800 and I've been very satisfied with it. There are some great prices right now on refurbed R1800s and even new R1900s. You shouldn't automatically overlook these, they are great printers. (I have to admit though, I am lusting after a new Epson 3880. I just love larger prints and would love to be able to print those 16x24 or x20s, what ever those larger sizes are.)
Good luck to you
Bill in WV