You were correct in your observation that your toilet looked like a modified oversize portable. Many years back, we had a Camper Van (1980 Dodge Maxi) with the hand pump version of the same toilet. In the left rear top corner, where you have the large white cap (water control valve?), we had a plunger pump, providing flush water. In our unit the flush water tank was twice the capacity of the internal holding tank, while the fresh (potable) water tank was about 6 Gallons (closer to capacity of internal holding tank). I interchanged the water feeds for a better match.
As to your continually running water problem: why not use a hand spray nozzle, as used by Thetford as part of their 'Water Saver' system. Personally, I think the Thetford attachment looks just like a standard kitchen pull-out sprayer, with a different bracket. BTW, kitchen sprayers are cheap, work well with city water pressure, have a standard 1/8" MIPT connection, and would probably do a better job of cleaning 'soil' off the bowl.
Note: I started to send this as a PM, but thought I would post it in the forum instead, in case there were others with the same problem.
I'm now looking at modifying our current Travel Trailer for recycled gray water toilet flush, using a standard kitchen sprayer.
The rationale behind this is: Gray water is generally the first to fill, and has soap in it, which would help to clean the bowl. Also, by not using fresh water to flush, you conserve that too.
Guess what I have as a pump for the gray/flush water? A Shur-Flo 205-103 pump, that dates back to about the early 1970's, and is still going strong!