As LittleInch suggested, readily available solenoid valves (even proportional ones) are not very good at controlling pressure directly. To a first approximation, what they control is flow (and, if you haven't got some sort of mechanism for maintaining a constant pressure drop across the valve, even that's a pretty poor approximation).
Here are two approaches you might try.
You could connect a single direction valve in series with a fixed orifice and tap off the pressure at the junction between them - the hydraulic analogue of a potentiometer. This approach can be made to give you something that responds very quickly, but is wasteful of energy.
The other is to use a bidirectional valve to charge or discharge an accumulator - much more analogous to your existing approach. Although the response is slower, pressure control will be smoother and you might even get away with bang-bang control rather than a proportional valve.
However, making your chosen valve control pressure is only the start of the job. You've still got to sort out how your valve arrangement is going to get a suitable and reliable supply of hydraulic fluid. It isn't enough just to clag in a small hydraulic pump, switch it on and then turn the tap on and off. To a first approximation (Peter will tell you how misleading this is, but for a first approximation, it will do), cheap hydraulic pumps are constant flow devices. That means that when your valves aren't demanding flow, you need to choose a suitable place for that constant flow to go (suitable doesn't mean out through a new hole in the side of the pump). In the motion-control world, it's common to find that the proportional control valve that does the simple job you wanted in the first place ends up supported by six or seven additional valves and a handful of orifices - without which the system will overheat, self-destruct, refuse to work, or some combination of the three.
If a handpump and bleed valve worked well enough in the first place, perhaps you could just replace the handpump with a small electrically driven pump with a check valve on the output and then only run the motor when you need to increase pressure. Replace the bleed valve with a solenoid valve and an orifice to limit the flow. Add an accumulator if you want to slow the process down.
A.