Tried this for like 20 times now. They amount of luck needed for this to work is just insane. That together with the worst AI ever makes it so freaking hard.
I quickly realized that you needed to start the game with a No CB war against either Ragusa or Serbia. I usually wait for the Alliance with Venice and then go for either of them. By that time Serbia has started attacked Bosnia, so attacking Ragusa usually is my starting point. Annex them and then claim Serbia and attack them. If you are fast you can vassal them and take over the war against Bosnia.
That usually goes well with a few quick tries. Also you need a decent Grandmaster, 3 or more is definitely desirable, almost required.
After that hell starts. As many other already said, you need to wait for Venice claim to end, otherwise they just break alliance. Around 1571 you can start the war against Cyprys. And then the worst kind of AI shows it's true form. Venice sometimes just sits around with their Navy, either at home or blockading some Mamluk province with all their Galleys. Giving Mamluks the opportunity to whipe out the rest of their navy. A reload usually fixes this but then Venice starts sending out suicide boats at Mamluks whiping out their own navy entirely.
Also Venice tends to send some small 8 stacks into Mamluk mainland into their 30-stacks.
And then there is the Ottomans. In all my games I have seen them attack Mamluks once. Even when they have the Levant quest they just wipe out small nations and tends to expand north into Serbia. Your only chance is to wait for Mamluks to attack some other nation using all their manpower, that happens once per game at max.
I did try to start as Cyprys and managed to get alliance with Aragon since you can Royal Marriage. But Aragon won't join any war against Mamluks because of Distant War. Also allied France as The Knights but same there, distant war and won't join. So your only ally that will actually ever help you against the Mamluks are Venice.
I did see someone recommending to vassal Byzantine. I never did that since I thought that would just get Ottomans up my ass. But I will try that since they are the only vassal you can have that has a navy to help out.
So my next try will probably be to attack Serbia when you are attacking Bosnia. Using Venice to vassal both Byzantine and Serbia. By the time you are done with Byzantine, Serbia has probably already wiped out Bosnia, giving you 2 strong vassals, one with a decent fleet. I will probably take Zeta to get closer to Austria and Poland to get an alliance. Then wait for 1471 when Venice's claim ends. By that time maybe diplo annex Serbia if possible.
By the way, what approach do you have with the conversion? I have always taken Religious Ideas and then rushed Missionary Strength. But I saw someone instead rushing Admin tech 8 where you get better missionary strength. What is the best approach there?