I think your primary obstacle will be finding a company offering luggage forwarding. Not very common in the Alps (most people in your situation would only have day pack and rely on the catering and sleeping accommodation of huts and/or hotels.
On suite private rooms high in the Alps are very exceptional, you’ll thus have to find a route whereby you descend (almost) every night back to a valley (read: a village with hotel accommodation). Again, not a very common approach in the Alps (most people with similar requirements as you have would rather stay in the same hotel and make day hikes in the area).
Have a look at – a part of – the Chamonix–Zermatt haute route aka high route (the summer high route that is, not the winter high route), and consider walking a section of it. E.g. start in the morning from Trient, Switzerland (no hotel in Trient itself) eastwards. If finishing in Zinal (or walking the valley route to Zermatt), if using the cable lift form Le Chable/Verbier and if using the alternative leg via the village of Grimentz, only one night needs to be spend in a hut (= Cabane de Prafleuri).
Plenty of trip reports online. You can most probably book all hotels & hut(s) via email/online. Don’t know of a company offering luggage transfer. Thus consider hiking with a lightweight day pack – plenty of catering options on route – and sending your luggage – wrapped in cellophane – with fresh clothes and other goodies to your hotel in Zermatt/Zinal (Geneva airport has a post office – obviously check opening hours & dates).
Get the ‘Swiss Transfer Ticket’, allowing a return trip: Geneva <-> Zermatt. On route to Zermatt, get off at Martigny and take the bus with destination Trient.
More info: get the Cicerone guidebook (can be purchased as an ebook).
The TMB is, together with the Tour Vanoise, most probably the most popular (read: busy) walking route in the Alps. Personally, I would avoid it at all cost during high season.