We had exactly this problem on a twin (MD80) in GVA - RW 23.
A tyre blew as we rotated but the first cockpit indication we had was a jolt in yaw and popping left engine. Immediate actions for engine failure on take off were Gear Up and check TO thrust set on the live engine - the popping engine ran OK in flight idle, where we had set thrust.
Those who know GVA know that the single engine climb out is terrain limited and exactly defined. With the gear up we just made the required min alt for the left turn.
After gear retraction we had a red on the left main, so taken together with the engine problem I had the first thought that a tyre burst could have occurred.
The popping engine was caused be the L spray deflector having been torn off and entered the L intake, where it sat over the static vanes and restricted the flow but we didn’t know that until subsequent trouble shooting in the PAS hold and coordination with the cabin chief, who could see the rubber through a pax window. There was also other serious damage, so the story goes on but the only relevant bit here is that the gear had to be retracted in order to get over the hill.
If you have this happen somewhere there is no single engine problem and you are pretty sure that a tyre has burst, then don’t retract.
Problem is, will you know in time to make that decision?