A wrist rocket to handheld is going to change things, generally the handheld brings your shooting hand forward a little more, it depends on how you have it setup. I'm assuming your anchor changes between the two. Kinda asking a lot to bounce between the two in my opinion. The rotation of your hand in the picture is going to be rotating your loop more than I imagine an index finger release will be, this can cause a few different things to happen. One of those things is increasing face pressure.
Another thing is if your draw length is perfect with the index (wrist strap) it's likely you are a little compressed with the handheld. If this is the case it might be causing you to pitch the bow a little much on the release. A lot of times it will cause you to bend your elbow a little more and in your release instead of the bow moving forward it actually goes to the left and forward, with a right handed shooter. From the picture you don't look too compressed in my opinion tho.
I don't ever expect different releases to tune the same. Just a lot of little differences.