Cost is going to be your biggest obstacle. Yes, it can be done. I have experience working with hydraulic drive systems and can without a doubt tell you it will cost far more than $1000. Your Hurth transmission will seem like a bargain compared to the cost you will incur with a properly designed system.
This isn't anywhere close to being as simple as a hydraulic drive system for a bow or stern thruster. Those systems are not designed for continuous use; nor do they have much horsepower. When you need a system rated for continuous use in a power intensive application as you describe, it becomes increasingly complex, hence increasingly costly.
You will need to have a cooling
system designed for such an apparatus. This cooling system will need to handle the high pressure flows of most hydraulic systems (unless it is on the low pressure end, which has disadvantages associated with it).
Once this system is designed, purchased/fabricated, installed, and tested...then comes maintenance
. That's the fun part of working with hydraulic systems. Every single
point of connection will need to be constantly monitored in the initial stages of operation. And in a system like the one you propose, there will be many many many points of connection. The oil needs to be changed on a regular basis...and perhaps more often than on a transmission. There will be several filters that need to be replaced frequently.
The quick and dirty answer is "Yes. It can be done!" But since it is your boat and your project
, I would suggest you do a lot of research
beforehand. A good starting point is a book titled "Introduction to Fluid Power" by James Johnson. It will give you a good starting point in thinking about the advantages and disadvantages of such a system.