The existence of planar distance preserver?
Many years later, it looks like OP has finally answered his own question: Near-Optimal Distance Emulator for Planar Graphs by Hsien-Chih Chang, Paweł Gawrychowski, Shay Mozes, and Oren Weimann was just posted on the arxiv.

The answer to the original question is yes: it is shown that $\widetilde{O}(\min\{t^2, \sqrt{tn}\})$ edges suffice to preserve distances between $|T| =: t$ terminals, which is optimal up to log factors.  In particular $\widetilde{O}(n^{3/4})$ edges suffice for the setting in the OP.  This preserver can also be computed in $\widetilde{O}(n)$ time; I would strongly suspect that the log factors in the size can be removed if we care only about existence and not computation time of the preserver, but I have not rigorously verified this.

(On a less formal note, I find this result really amazing.  Congrats!)