My earlier comment was not wrong, but in fact one can do something with Ricci flow, see Bamler-Kleiner Remark 1.3. The two metrics in question can be connected by some path of metrics, and one Ricci flows the whole path.

How would Ricci flow provide a proof? The metric has curvature 1 so it just shrinks...nothing actually happens.

This space definitely does not have a Heine-Borel property. The point is that you get compactness in a weaker topology given boundedness in a stronger topology.

The OP says $p,q\ge 0$ which includes $p=q=0$, i.e. the sup norm of $f$ itself. So this is a norm.

This is a standard exercise in advanced calculus. You apply the Arzela-Ascoli theorem to the sequence and its derivatives. The main ideas are in Chapter 5 of Jost's postmodern analysis book.

Here's a version that gives what I claim. Suppose $X$ is compact and let $\theta_1,\dotsc,\theta_N$ be a partition of unity relative to some system of charts that covers $X$. Then define $$\|u\|_{C^k}=\sum_{i=1}^N\sum_{|\alpha|\le k}\sup_X |D^\alpha(\theta_i u)|,$$ where $D^\alpha$ is the $\alpha$-th multiderivative in the chart associated to $\theta_i$. This norm should be equivalent (in the sense of norms) to what you wrote. It has the desired compactness property, so your norm does too.

Is this not (equivalent to) the standard $C^k$ norm (at least locally, in charts)? If so, this is just the Arzela-Ascoli theorem.

@DCM If I recall correctly, this is what is done in Gilbarg-Trudinger. The existence theorem in Schauder theory is by continuity, so one needs to start with an existence theorem for the Laplacian. The best way to get this seems to be to pass through the $L^2$ theory.