Proof technique in complexity theory
You can indeed assume that you know a polynomial bound on the time it takes to compute a polytime function. This is an example of a non-constructive argument. 

The down side is that you won't be able to construct a computable function transforming polytime function of one model to the other. In some situations (say a diagonal argument) this is needed, and then the trick is to assume that programs are clocked, i.e. come with their own time bounds. This makes sense in that application since we can enumerate all polytime functions as computed by polynomially clocked machines. In your case, however, you don't need to worry about such issues.