Just curious, if we ignore the varnish issue (I'm generally not a big fan of varnish either, at least not on lighter woods), how does the press work?
I followed a similar design, but I did it very quickly and so everything isn't perfectly square - I think I did mine from conception to completion in 2 hours. It's pretty sloppy. Anyway, my problem is the top "tray" tends to bind... It's not such an issue when I have weight on it as long as I center the weight, however when I'm just putting the tray on or trying to take it off, it can be a serious PITA. I'm pretty sure it's because I didn't bother to square up my dowels.
By the way, I'm not sure what diameter dowels you used, but I used 1" dowels and they happen to be exactly the size of quite a few weights... I had a bunch of weights in my basement and I could easily add weight as I needed it, and with the 4 dowel design the weight is evenly distributed (which helps keep my top from binding).