That does not look like enough room for R38 which is 12 inches thick. Whatever you use since you have no eave vents you are better of filling the cavity and putting a good vapor barrier like 4 mil poly rather than depending on the kraft facing so you can be confident of stopping any moisture.

Closed cell spray foam is impervious to water and you could get your R38 in less space but it is expensive.

Depending on how the porch roof relates to the rest of the home's roof, you may be able to install rigid insulation above the roof deck to attain the code-required R-value. That would allow you to maintain the look of the interior. Here's a link to additional information to get you started:

https://basc.pnnl.gov/resource-guides/above-deck-rigid-foam-insulation-existing-roofs

You don't have enough room for effective fiberglass/cellulose batt insulation but 3-4" of closed cell foam or closed cell foam board will work for you (based on the pictures you have provided). Having said that, I'd tackle the main house first!