Its not really wise to have differant makes or miss match of differant treads on the car/axle..of course you could ensure both your Michelin tyres on the front - and the two others are on the back axle..ensuring the front driving wheels have both the same make/tread pattern - as thats where most of the drive to road surface/steering of course and braking effort is on the car...If you do that by putting the two michelin's on the front and the steering issue is solved then you know you need to get the back ones done to match or that Dunlop tyre changed...doesnt have to be michelin..as long as the two tyres on each axle is the same brand/tread pattern..I do hope this helps
On your tyre the profile is the 2 digit number on the side wall -
so ( 123/45 R67 ) as an example . then the 45 will be the profile of the tyre..
But the treads surface areas or pattern shouldn't be differant or all differant on 4 wheels you want to avoid that..its possible that tyre may have suffered damage due to potholes or kirb damage..only way to know would be to get it checked out at a tyre centre to be honest..or maybe try as I suggested above and put both Michelin tyres on the front and see if the problem of pulling is solved