Since, You know that it is either Brown Turkey or Black mission, both are 2 crops varieties, Both are known that easy to get figs.
Therefore, grafting is a bad idea, highly likely waste your time.
Boyer is right, I think that trying to Fertilizing them is better idea to get figs.
If you don't mind my story, I live in PNW with cooler climate, I am newbie and want to enjoy figs this year, I bought 8 nursery fig trees and planted them in ground at March, Due to micro climate of colder in my garden, none of them bearing figlets while same fig trees at nursery, full of figs, in order to get figs, I applied higher dosage of fertilizer, 24-7-16 or 10-10-10 or Tomatoes fertilizer leftover from other garden projects, After 2 weeks, 50% of them have figs, after 5 weeks, all of my trees had figs, My Co-worker at work, She loves fig, She bought 5 fig trees from Costco at spring time and planted them in ground, none of them has fig, She is so upset.
I did plant 2 fig trees in shaded area between 2 houses, it has about 7 hours SUN, they do bearing figs but later and less figs, But problem here is that, Due to cooler climate here, Figs take longer to be ripen before fall/winter.
If I were you, I would apply 1 bag of big box store whatever "manure" to your fig tree, Then at spring time, If your tree leaves out and you don't see figlets, you might apply some fertilizer to soil rightaway.
Prune them is a good idea to get figs too, It might scare the tree to bear figs or it reduces tree Canopy, so extra leftover nutrition to support branches are used to get figs.