Ok, I've completed my modifications, and they're now working 100%.
This first picture is the "bridge" I built, v1. The second shows it installed under the sink. The third picture (sorry for the poor quality, cell phone) shows it installed with the third cut-off valve installed in the further horizontal pipe in the bridge. This successfully stops the high pressure water from bleeding into the low pressure side.
Now, I have 3 modes of operation:
1) all valves vertical. This is "normal" operation, and it works just as the default plumbing does. The pump takes water from the tank (if no city line in use). Otherwise, the city port works as expected. Like this, there is no functional difference from an unmodified TM. Most of the time the valves will be in this position.
2) all valves horizontal. This is the "tank refill" mode. I can hook up a hose to the city port and drop the end into any container of water, and the TM pump will suck the water up the hose and refill the fresh water tank. So, I'll hook up the hose first, then go inside and twist the valves, turn on the pump, wait until it's done, and then shut it off and move the valves back to position (1).
3) if I just move the lower-right valve (easiest to access, too) to horizontal and leave the other two in their default vertical position. This is Fill-Up mode, and I do this before I close up to leave my home and go camping. I always travel completely dry, to save on weight, and fill up at the campsite. I've added a female coupler to the high-pressure drain only. In this mode, I can hook up the hose to that coupler and fill both the high pressure side, the hot water tank, and the fresh water tank, without needing to move the hose, reconnect, or anything else. I start with all the drains open except the fresh water tank drain. As water starts coming out of the high-pressure drains (cold first, then hot) I close those. Finally, when the fresh water tank is full, I've moved my vent tube to the underside of the TM, so water will come out there. That's my cue to turn the water off, close the last drain, and disconnect. When I am ready to use the water system, I just turn that one valve under the sink back to vertical and turn on the pump. Since the high pressure side is already full of water, it takes only moments to reach pressure, and then the pump shuts off as expected. This is a far cry from my old method, of popping open both shells, filling the tank through the port, turning on the pump half-way to let the heater fill, waiting until water comes out of the faucet, turn off the pump, then watching until the tank was full, disconnect, close up. This way is much easier and faster.
Rerouting the vent tube was simple: Mine was already disconnected from the vent near the fill port (it never was connected, for whatever reason). It's 3/8" ID vinyl tubing, so I got a length of that from Lowes and a 3/8 to 3/8 barbed connector. On my TM the high pressure drain pipe comes through an opening that is large enough to also contain the 3/8 tube. There's some foam filler in there, but it was easy enough to use a fat hex wrench (anything will do, really) and poke out a hole, and feed the tube in from the bottom. After that, it was simplicity to hook it up internally to the original vent tube. I didn't bother with hose clamps on the barbs, because this will always be unpressurized overflow water, so there's no need. You can see the vent/drain tube in the second picture, looping underneath the pex, with the barb just visible on the left side.
As to installation of the bridge, I needed to remove the drain pipe from the sink for access (seen in pic#2), then using a ratcheting pipe cutter, I cut the pex pipe coming from the city outlet, leaving 1.5" at each end and removing the middle 4.5", as this was the width of the bridge on that side. Doing this meant I could reconnect all of the rest of the standard piping as-is, just replacing that part of that one pex pipe with right side of the bridge. The other side of the bridge connects on both ends to the braided flex tube from the water tank to the pump. I cut that tube fairly close to the pump end, leaving enough to be able to make the connection, but cutting it closely enough that the tube goes from bridge to pump without any kinks or bends. I needed to trim it twice for the fit to be perfect. The other end just hooks on, because it goes down to the tank. I was worried there might be a corner bend in the tubing this way, but it took on a gentle curve with the underlying (pre-existing) pex tubing offering something of a support. To add that last cut-off valve, I had to carefully measure out the middle of that last pipe, clip it with the cutters, and then insert the valve. Retrospectively, I could have installed that valve during the construction of the bridge pre-install, and it would have been a whole lot easier. The pex that goes to the city outlet will unscrew from there, so that helps a lot in the installation, but beware that there is a gasket in there that needs to be reseated properly. I didn't, the first time, and water sprayed all over the place when I pressurized the system for testing. It made a complete mess and caused my home GFI to pop for a full day until everything dried out.
I chose to use flair-it fittings for everything, because they work without tools, they're light weight, the valves are full-flow (so no restriction on water flow over the standard installation), and they're a lot less expensive than sharkbites or similar products. They're all plastic, so there's no issue with brass or lead or anything else. Plus, my dealings with the company were really great, and they helped me troubleshoot the pressure/leak issue I was having--which was a result of the design of the valves, not a defect. My application is, shall we say, not the way they designed these valves to be used, so we needed a little extra to make it work. On the other hand, the connections require you to really push them onto the pex, and sometimes that was a bit difficult to do, depending on the direction of the "push" and working in such a tight space. Knowing what I know now, there was only one connection that would have been a little tricky, as the rest could have been done before installing it under the sink, which would have been much easier. Just build the whole thing, then put it in. Nonetheless, those connections never leaked -- only the screw on one at the city port, because of the gasket problem I mentioned above.
I'm quite pleased with how this all turned out, especially now that I no longer need to worry about running out of water and refilling the tank. I looked at the deck plate solution, but trying to hold even a 5 gallon jug inside the TM, possibly with or without the twins "helping", and having no support, worrying about slipping and spilling water all over the floor...maybe it's an overreaction, but with this mod it's now a non-issue. Yes, I probably could have let it gravity-drain into the tank, too, with some tubing and a deck plate...but knowing what my counters look like after 3-4 days out, I just don't want to have to deal with it. Now I don't! No lifting, just connect a hose and push a button.
I want to take this moment to thank everyone who worked on thinking this all through, especially Paul and Dave, I would never have come up with this myself!