If and when you get to staging, you can bring multiple stages of trimix and minimally impact your backgas. In other words, start with the stage, switch to backgas when you empty the stage, then complete your deco. On subsequent dives, start with a stage, or if you're out of stages, then start on the backgas.
The backgas could last multiple dives before you drop below min gas requirements. You also get to dive trimix on all dives rather than dealing with a diluted version. If a dive happened to be within nitrox range, you could even stage with nitrox and barely touch your backgas except on gas switches.
I know that may not meet your objective to save on trimix, but you avoid diluting backgas for your second dive and avoid the validated gas switch. This method would also reduce your LDS visits, as a couple of stages and a set of doubles might last 4 dives, requiring 4 trimix cylinder fills, but could be done in one visit. Doing it your proposed way would still require two doubles trimix fills for 4 dives (4 fills), plus your travel stage fill and the doubles top-off fill after each first dive. That would require an LDS visit after the first two dives to fill the doubles with trimix again.
This trimix stage technique is what I use to help extend the trimix fill in my doubles, otherwise I'd end up having to dump the remainder after two dives. Banked trimix is not an option.
However, if you haven't used more than one deco bottle or stage bottle it is more complicated and needs practice - there's more gas switching, clipping on and off, stowing, and deploying with less space.