Well my reason didn't pan out because of the time between Andromeda galaxy collision actually happening. But still the knowledge of such collisions occurring I would imagine to be the prime reason for Reapers to go over to another galaxy so its Chaos doesn't disrupt THE CYCLE!!!!
Ever since Andromeda was announced I did fancy the idea that the Reapers went to other galaxies or maybe a show they had tried to take control of other galaxies. Since we don't want Reapers again I thought well why not Reaper Carcass for Andromeda and thats why we are mainly seen as a threat. The last intergalactic visitors tried to take over.
There's no point in going in another galaxy that won't reach ours before 4 billions years for this. That's at least 8000 cycles away. 8000! It's like if I said, as an European, that I'm afraid of the mess some random guy in America might eventually bring when all the continents will have floated all the way back together...
I think we will be simply seen as refugees. Some will try and welcome us, some will resent us for taking the ressources they themselves covet, and some will take advantage of us. Good samaritans, nationalists and slavers/organised criminals, oh, and a vast majority more or less indifferent as long as they aren't directly touched. Really, nothing different than what's happening here on Earth each time entire populations have to leave their lands to survive. No need for Reapers to find ourselves in such situation.
I can't totally dismiss the possibility of your scenario happening in MEA, but as I see it, it would raise more problems than it's worth. To beat our cycle, Reapers needed to engage all of their units. Andromeda is bigger and has more stars than the MW, and the Reapers have no control over how Andromedan civilisations and techs evolved. Therefore, they would need even more units than what they currently have available if they hoped to take control of it with an acceptable number of losses on their parts (the more Reapers they lose, the more harvested civilisations are wiped out, which is in complete contradiction with their mandate). So if they were beaten so badly in Andromeda, shouldn't it have affected their current numbers in our cycle (any Andromeda invasion couldn't have happened in the first couple of cycles, they just weren't enough and Andromeda was even further away)? Shouldn't it have prompted the search for a new solution earlier? Why didn't the triumphing Andromeda species invade the MW, following the Reapers? As I said, I can't dismiss your scenario entirely. It could happen, but it raises a lot of new problems. If they are going for a reboot, I'd rather, and that is a personal preference, that they start with as clean a slate as possible.