Although not familiar with PSE, I suspect it has no softproof functionality (simulate print output on screen) one possible way forward is the following:
1) if not yet done: calibrate your screen with one of the many solution available so that your screen displays colors somewhat as expected. This is a mandatory step before moving further!
2) create or get a custom profile created for your printer and the papers you use.
After creating the printer/paper profile I can softproof in the regular PS CS and get some idea about how the print will look like. The softproof shows (the same) differences I always have to correct to get it back at the original: a bit darker, a tad more contrast in the shadows, slight shift in colorbalance in the highlights)
3) So I went back into the software that created the profile from the measurements and dialed the correction into the (new) profile as this software allows to do this. It took a few iterations to nail the settings.
Now the non-softproof versus the softproof hardly differ and most importent the actual print is also okay, at least the differences are below my level of concern (YMMV ofcourse) and I only now do reprints if after some time I think that my creative decisions around the color need improvement. I almost do no reprints because of screen-print mismatches you suffer from currently.
Yes, the softproof and print still shows differences but that is because of the difference in paper/screen media, gamut differences between camera, screen and printer plus the effect of the chosen rendering intent how to handle the gamuts from device to device.
To summarize: get a custom profile created but one that will incorporate the standard adjustments you do to get a best possible screen-print match, that may not be possible with a remote profiling solution, you probably need someone to come on site or buy your own profiling hardware/software.
I don't know if PSE would be my choice of tool, I understand you might find the regular PS too expensive, but keep an eye open on Adobe Lightroom, although Lightroom still does not support softproofing, it's a much requested feature by users, I would not be surprised to see it in the next version. And that would also be one of the last features needed for me to switch over the much more friendly priced Lightroom......