Flashole: Your Big Bore 444 will shoot any jacketted or gas checked bullet with all the accuracy you need. There is no trick to the micro-groove barrel except that it doesn't shoot plain based cast bullets worth a dang. Jacketted and gas-checks are an entirely different story. Beartooth Bullets has conducted an incredible amount of research on this issue with Marlin rifles in 444, using the same barrel and twist rate found on some of the Winchester Big Bore 444s, and has obtained incredible accuracy with their gas-checked bullets all the way up to 410 grains. I purchased some of those bullets and they are monsters. Harpy, who also posts on this forum makes his own jacketted bullets for the 444, all the way up to 410 grains and they are beautiful to see. The accuracy he has obtained from his Timber Carbine with a 1:12 twist is 'minute of angle' and phenomenal.
The setup you have, from my perspective is the perfect woods gun for anything on this continent. Yours will shoot any 240 or 265 factory load very, very well - maybe even better than you can with open sights. And speaking of sights, an Ashley Ghost Ring setup will serve you well and improve your sighting.
The micro-groove barrel will not lead any worse than any other barrel if you stick with jacketted or gas-checked bullets, period. I think the micro-groove rifling is a bit too close together to allow plain based bullets to take the rifling well enough to give you any decent accuracy. And since you have a new barrel/rifle, you may want to throw a whole bunch of jacketted stuff downrange to help break it in. Also, a Tornado brush will help you keep that barrel clean.
Actually, I do not know why Winchester dropped the Big Bore 444 from their inventory but I doubt it was due to any accuracy problems. Gun Tests Magazine and Guns and Ammo reported on a comparison between the longer barrrelled Marlin rifle in 444 and the Timber Carbine (same twist as yours). Both magazines tested the guns with factory loads, including some from Garret and Buffalo Bore. All loads wore a jacket or gas-check and all performed with better than expected accuracy. The only comment about either of the rifles tested was that the Timber Carbine was about as light a weight of rifle (6 3/4 lbs) that you would want to use with this caliber but beyond that they gave no discredit or had complaint with the accuracy from that 1:38 micro-groove twist. Anything quicker in twist just helps improve the accuracy, as Harpy's results demonstrate, but unless you are into 'minute of angle' lever actions your new Winnie should serve you very well for a very long time and group under 3" at 100 yds with the factory open sights. With improved sights your groups could easily shrink to 2", which is excellent for open sights on a lever action.
Oh, you also mentioned that the slow twist won't stabalize heavy bullets - this is not correct. The 1:38 twist was first used, I believe, in the Trapdoor Springfields firing the 45-70, which used heavy bullets with great accuracy. You should not worry about that. Somebody told you wrong. The only issue about the slugs you use is whether they are jacketted/gas checked or plain based. If you had a 6 groove barrel in 1:38 twist you might be able to shoot plain based cast bullets better than with a 12 groove micro-groove barrel but with the advent and availability of new heavy bullets for the 444 in gaschecked configuration, there isn't anything out there now that you can't shoot from that barrel with the kind of accuracy that will make you very happy.
Flash, ya dun guud with yor neu Winnie. Get to know her and have fun. From the bench she will be a butt kickin' mule but when you drop the hammer on a running whitetail, some nasty assed old boar or a bear who has gotten too close you won't even notice it. This is Mikey and I hope this helps.