Don't worry about the gesso, it wouldn't be needed here, the easiest way to make up a sizing in this case would be to use thinned carpenters glue, most of them are PVA, thin at a 6:1 ratio (water:glue) and simply brush on, after the first coat is COMPLETELY dry apply a second coat. If you are using any modern paint you will have no problems with this method at all.

The method if you were to use historic materials is a bit more complex, but not all that different. Using hot hide glue and working with milk paints is just a slightly different skill set than most of us have.

P.S. a PVA glue size cleans up easily with plain white vinegar.

Medevial knightly shields usually covered with parchment, at least the preserved ones. However, these are usually very ornate affairs, with raised reliefs and such. (which is probably the reason they have been preserved, while the plainer ones where simply discarded.

In the Norwegian laws, the minimum demand for a shield is "three iron fittings running across." In the later (13th c) laws, that go into greater detail, the impression is that the iron fitings are used for shields without a front cover.