Right, so not the Canal of the Pharoahs, but a legit Suez Canal.
That is still a better canal, but damn if that isn't at risk of bankrupting whoever built it. Perhaps the Mamluks partner with Timur. Perhaps during the 3rd reign of an-Nasir Muhammed, we're looking at 30k people at any one time, which is basically an army in its own right.
The motivation is a little more odd - I suppose if the Mamluks start considering expanding trade with Europe, or a scheme to undermine the trade routes of the east. (Effectively a White Elephant project until its complete)
There are other benefits, when built it becomes without a doubt the fastest route to the Hajj from Africa, Egypt, Syria, etc. The major benefit is that they could use the canal to increase their revenues and the quantity of trade they receive through Venice.
Sourcing the slaves could be anywhere from prisoners of war, slave purchases from West Africa via North Africa, Ethiopian slaves (in fact, the Mamluks could even send an army to what is now Eritrea to capture slaves and take more control over the straits) - all of which can work to build the canal, however slowly it may be.
But we've finished in in the 1300s. Lets go late 1300s. We have a Mamluks that has probably appeared weaker, suddenly more powerful than its ever been if it completes the project. It can build a navy that can operate in the Red Sea and Med, and is able to cut Persia (and thus also the Turks) out of the Silk Road and Indian trade because its quicker, cheaper, and easier to trade with the Mamluks. Strategic concerns suddenly include places like Hormuz, and overseas in the Maldives and Ceylon.
Basically it creates a new Mamluk Golden Age when it historically died. Or if someone still kills it, makes them all the more dangerous. It also makes Venice more dangerous - it is the major trade partner, it could very well be paying the fees to go trade with India.