Sorry, work a bit of a pain this morning - overslept.
Peace to Persia will probably be enough to clear the war weariness. I think we have lost three or four units to each of the Romans and Americans. Peace with them is just to get our heads clear again, stop distracting ourselves and allow the military to concentrate on building up and preparing for the next war. Which as you say is probably Portugal, and will need more troops.
We should be able to get enough smiles from luxury trading to keep our lux slider at 0%.
Mm, techs. We somehow got a bit turned around here, with the lack of a decent Copernicus prebuild being at the root of the problem. I tried putting Joan's current tech situation into
Ambiorix's AI research guesser and it predicts relatively similar probabilities that Joan is doing Navigation (least likely), Chemistry, Music, Printing or Banking (most likely).
So our best path would have been get Astro, trade it for Guns as soon as we could, go Chemistry, and swap that for Joan's Banking. Indeed, given that the size of our Copernicus prebuild limited the date of the Astro<>Guns trade, I think Taxpayer was actually correct to start self research on Guns!

The other option to tide us over that period would have been Printing.
Anyway, here we are due to finish Banking soon, and we need to research something else. I think we should sell Banking to Joan right away, (assuming she doesn't beat us to it of course) and start Chemistry. The model predicts that Joan is more likely to start some optional tech than go for Chemistry, so to keep the pace going we must do it ourselves, and maybe we want to buy that optional for use in the tech trick.
Indeed, the only remaining medieval tech with a significant AI weighting is Magnetism, so we should try to park Joan on that tech as soon as possible? It would seem that the best chance to do another 2nd tier tech trick would be (1) research Chemistry, trade it to Joan, (2) research Physics, trade it to Joan, (3) research Gravity (shall we take Newton for Constan? I would vote yes), (4) research an optional or two, and start Metallurgy, (5) buy Magnetism from Joan, (6) finish Metallurgy and go industrial, (7) trade merrily on the interturn and get Fascism free!