Getting the desired force measurement from strain gauge readings in the manner suggested is probably not going to be a trivial matter.
For one, strain gauges measure strain (elongation) rather than force; translating strain measurements into a force measurement requires accurate knowledge of the mechanical properties of the system, which in the case of an item like that pictured might not be as well-defined as one might imagine. Aside from the matter of measured strains being dependent on precisely where on the structural member one chooses to take the measurement, you probably don’t have a very good handle on the precise dimensions of the structure, what it’s made of, the effects of any heat treatment (e.g. welding), work hardening, etc. All of which will factor into the relationship between force and measured strains.
Second, there are a lot more forces acting on those members than the ones indicated above. By virtue of being structurally conjoined, both the push rim and wheel rim will be taking up forces due to loading on the wheel, which of course would be dependent on the weight of the person in the chair, what direction they’re leaning, and probably even what their uncle ate for lunch last Thursday…
If the ultimate goal is to measure the tangential force applied to the pushrim, your best bet would probably be to investigate ways of modifying the structure so as to focus that force along a linear vector, allowing it it to be measured with a single, well-calibrated load cell or two. Otherwise, the risk of acute hair loss come data analysis time is likely to be quite high.