The properties of TPU and the sensitivity of your capacitive touch sensor and your desired "feel" of sensitivity will determine your maximum protector thickness. I don't know if phones have a sensitivity adjustment in software that would be relevant, and and EE or programming stack exchange answer would be suitable for that. You would have to be careful not to phrase it as a "use of consumer goods" question, which is forbidden, and you'd need some programming and phone hacking skills to get a useful answer, so you may wish to simply find a method to determine maximum thickness of your desired materials, such as simply applying normal thickness layers over each other to determine the thickness.
I've been completely satisfied with a thick, rigid bezel(far more effective than screen protector for most bend/drop situations, especially edge drop situations. For the odd time you drop your phone face down on a piece of ceramic or self-tapping wafer screw(hard object), if double thickness is not adequate for you, I would suggest an extremely thin regenerative wet install film for an inner layer, a sacrificial tempered glass protector over that to provide rigidity, and your TPU layer on top of that for abrasion resistance and shock absorption for the tempered glass. I believe this may even allow the inner screen to survive fairly hard hits on "magic" rocks and "immovable objects", like toilet ceramic or diamond, or firmly mounted pieces of hard metal. A thick TPU protector would survive hits better itself, and protect the screen better for many types of hit than a thinner protector of the same material, but is much too flexible to provide decent protection from hard point hits.
I'm pretty sure the adhesive need simply be optically clear, wetting to some degree and of course temporary and suitable to both materials.
Having decided that, an EE can only call around to chemical engineering companies and adhesive suppliers to see what might work, and in the industry that particular task would be rare for and EE as it would likely be done by a materials engineer, case designer or phone retailer/repair agent.
What adhesives have those properties, is indeed a product knowledge(shopping questions are not allowed although we typically give a few days for people to volunteer what we know offhand before inappropriate questions are migrated or deleted) or materials engineering question, which should be re-asked on the correct stack rather than migrated in this case, as this question now has an EE answer, to the degree that it is an EE question. The Engineering Beta(general engineering) stack exchange or physics stack would be the best places to ask that, as far as I know there is no chemical or materials engineering stack.