Yes, it does still make sense to gift him a tech now. Bettet late then never

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Instead of trading resourcea, you could maybe get him to pleased via 1 or 2 tech gifts and then beg the resource from him. It's very likely that he'll agree as it's the first time you ask, giving you 10 ensured peace.
It may also be, that he cannot even DoW you when being pleased, for that information check the Civ Illustrated #1 guide.
I tested the possible options a bit:
- Bribing Mansa against Monty didn't help. BurgerKing is still declaring war against me instead of Mansa.
- I have to give 2 techs to BK to bring him to +1 fair trade. He stays at cautious however and still declares war against me.
- I have to give 3 techs to BK to bring him to +3 fair trade (which drops to +2 after around 5 turns though). He stays cautious but doesn't declare war.
- And if I give him all 4 techs I get +4 fair trade but still cautious. But he doesn't declare war.
Wow, is that difficult! Anyway, I know already I don't have to care so much about his DoW because nothing happens. But that knowledge is kind of cheated.
But reading all your tips you seem to consider managing diplomatic relationships in the early phase as very important, right? I never really did to be honest, that's why I usually have a lot of cautious relationships (except when sharing a religion or not). The main reason is, I believe, that I don't give the AIs something without getting something back. So, I normally never gift and I even trade techs in most cases only when the beakers are almost on par. I have to change this habit and somehow get a grasp where the limit for gifting is. (Gifting 4 fat techs for the +4 fair trade and then even not at pleased looks kind of rip-off from the BurgerKing.

)
OK, just to finish diplomacy now and to focus on the war: I gift BK 3 techs now. And also a few techs to Gilgamesh because there are actually even more tensions with him than with BK. And then when Mansa has almost completed Construction I gift him that tech as well.
And if you say, that's crap, I go back again and replay. I have good practice in replaying now...
(BTW: BK doesn't DoW at pleased.)