Matt, a couple of observations:
First, SIL had a set like your first picture. Painted MDF. Did not last even though the two young-ins were 7 &9 yr old little girls. The beds were torn up within a couple months, because the MDF stretchers did not hold the fasteners when the 45 lb and 60 lb girls decided to bounce on them. I found out the girls were sleeping on the floor, so made some box frames out of 5/8 BCX pine ply that fixed things until new beds could be afforded (never buy rent-to-own). If going with that design, use multi-layered ply (cheapest being something like Arauco from Lowes at $35 a sheet but lousy to stain and finish due to blotching and the surface plys like to lift when sanding) to Birch or oak ply at $45 a sheet. IMHO MDF is cheap but will not last long enough to make it worthwhile.
Your second picture and Dan's picture seem to be better designs. Use minimum SYP pine for the uprights and bed rails. Ash would be plenty strong enough. I have never used Aspen so do not know, but poplar is too weak, especially for the bed rails. Besides, poplar is very difficult to match any other wood when staining, finishing.
Oak ply matches well with ash for the flat panels, and will be stable, so wood movement does not have to be a factor in the design. If pine (cheaper) for the uprights and stretchers, 5 or seven ply ACX (BCX might work as well but you'll have to pick through the stack) pine ply will work and has thick enough outer plys that they clean up well run through a planer to get a smooth surface.
In your neck of the woods, Douglas fir may be more available and cheaper. It is one of the firs that has the structural strength if dried properly (can be used for load-bearing construction). I have not used it, though, so do not know how it machines, stains or finishes.
Around here, if I made it and my name was on it, and on a restricted budget, I would use ash for the frames, oak ply for the panels, poplar for any drawer carcasses. Drawer fronts could either be ash or edge banded oak ply. If too expensive, I would also make it out of pine (SYP, not SPF), but that will mean more scrap to get good boards, as well as more of your time machining the lumber, so will cut into the price reduction.
If that doesn't meet their price range, I would tell them you are not going to make something too cheap as it will fail, and they might need to look elsewhere. If you make it cheap and it fails, you will be the one that is blamed, not their unwillingness to cough up for decent material. (DAMHIKT)
Go
PS. My aversion to SPF is due to the quality of the stuff we get here. Beings you are closer to Douglas fir country, the SPF you get there may be of better quality. You may want to grab a couple of 2 x 4's from the borg, mill them to 3/4, and see how they work as for fastener holding,etc. If a #10 screw can be ripped out fairly easily with a claw hammer, it probably is too soft for the main framework.
