I'm unfamiliar with dealing with the winter lanterns with a gun. Are they susceptible to blood damage?
not particularly compared to other forms of damage. it's a combination of a few factors.
the sheer advantage of attacking at range. the winter latern can't actually attack you or deal any damage to you if you don't get close. and if you kill it fast enough
the WORST thing that can happen is that your frenzy meter builds up and you take 90% damage... but then the winter lantern is dead, and you just heal yourself.
most people do not play the game with fully upgraded guns, due to the shortage of upgrade materials. a +6 gun with mediocre damage is going to take like 20-25 shots to kill a winter latern. obviously, when trying this out, you're going to "learn" that using a gun doesn't work against them, which gives even more reason not to upgrade your gun since you think it's useless. but remember, most weapons spike in power from +7 to +9, and guns are no different.
compounding this issue, a lot of people use their gun to slot the health regeneration rune. another thing that causes their gun to do much less damage.
basically you just have to actually approach the fight with a +9 or +10 gun, appropriate for your build, with a suitable blood gem slotted in. this will give you the damage needed to kill a winter lantern very quickly with a gun. and if you can do it fast enough, they have a 0% chance of killing you. at worst, you'll spend 1 blood vial and maybe a handful of bullets per winter lantern... to completely eliminate the risk of trying to fight them in close quarters. for me, this is always a worthwhile trade.
(edit: also make sure to use bone marrow ash on your first shot. the hunter pistol, repeating pistol, and evelyn all get 2.55x damage from BMA buffed shots. these would be the guns you should be choosing between, depending on your build. and this will absolutely chunk their health with the opening shot. it might even be worth reloading and firing a few BMA shots rather than firing a bunch of small shots, depending on how fast you can kill them)