The FL still did its job.
Yep. And sometimes the greater wisdom is to accept the small inconvenience posed by enemy units doing their job rather than morph your attack plan to avoid it.
In this case, you might actually be hurting your chances more by going to great lengths to avoid being slowed down by the MG than if you'd just let the MG have its moment and slow you down. I rarely have the units to burn trying to run up and down a FL trying to malf it. If you wind up diverting a tank and an Assault Engineer (smoke) squad to the area to help you deal with the MG, ok, that may just be good combined arms tactics, but at some point you can be spending more effort than it's worth. You've got concealment and winter camo, and a German HMG is staring you down. It may be better wisdom to lower your expectations to "let's just get most of our guys across the street unbroken" rather than "let's not let this HMG slow down the attack".
From the defender's perspective, he'd probably like nothing better than to get a bunch of 4-1 shots against assault moving troops, and it does indeed sound like a good deal for him. Your best bet may be to settle for denying him that good opportunity, trading units for time (your slowed attack). Hopefully the the HMG firelane keeps it from being used against you elsewhere - can you push on other fronts?
Still, I get it, you don't want to just concede the advance. Some probing is almost always worth it, because hey, maybe the MG does malf or maybe the FL shot misses entirely or maybe your guys do roll well on their MCs. You don't know until you try. Just keep an eye on the overall cost-benefit ratio here, and be willing to step away from the table should things start to get pear-shaped.
Is Dash available, given the terrain?