It's not clear to me this is a problem instead of an opportunity. After all, look at how Texas received a lot of German settlers.
I'm also not so sure that those American settlers who stream west will remain loyal to America. It didn't seem to pan out that way in Canada, after all.
'A lot' is an exaggeration. At no time in the history of TX (during it's Mexican dept. existence) did it have enough people to qualify as a full fledged Mexican state with representation. In fact, Santa Anna considered merging TX with Coahuilla to make a single large and thinly populated Mexican state. And TX was a garden state compared to LA... the latter was limited to one city and a couple of towns; everything north was wilderness. France could probably attract some settlers there... just not a lot. Frankly, if Europeans are going to emigrate overseas, America will be priority #1... because it's a big settled place with towns and land and jobs.
As for those Americans staying loyal to America... okay, it didn't pan out in Canada, but you remember what happened to TX and CA and HA, don't you..