I haven't read/heard anything that would suggest one manufacturer or the others' rifles "liking" or "disliking" Lead Free Bullets. I CAN tell You, however, that I use Barnes Bullets in my Browning A-Bolt .270 Winchester.
I've OFTEN read about quite a few Handloaders having problems getting the 130 grain Barnes Triple Shock X (TSX) or Tipped Triple Shock X (TTSX) (in .270 caliber) to shoot accurately. I use the 110 grain TTSX in my .270 with a powder charge on the lower end of published data using 110 grain bullets.
Say, for instance, You generally use 180 grain bullets in a .30 caliber cartridge. Try the 165 grain or 150 grain Barnes, instead of the 180's. Use a bullet a little lighter.
When my .270 had the factory 22" barrel, it shot the 110 gr. TTSX very accurately. I've rebarreled that rifle with a Douglas Air Gauge #4 Stainless 26". The exact same loads shoot very accurately in the new chamber/throat/bore. I load the 110 grain TTSX in .270 Winchester with 54.0 grains AA4350, and seated to 3.225". I did NOTHING in the way of "how far off the lands". I used the Cartridge Overall Length that's listed for the MRX bullets in the Barnes Manual #4, since that manual does NOT have data for the TTSX, and the MRX bullets have the same Ogive and Polymer Tip as the TTSX. Also, if You notice, the factory ammo that's loaded with Barnes TTSX, have the bullets seated to almost the exact same depth as I seat the TTSX to (the end/rim of the case neck is "over" the forward-most groove of the bullet shank), and I've read Rave Reviews about their accuracy...despite "how far off the lands".
One thing I would keep in mind, in looking for a rifle that You'll shoot Lead Free bullets in, is some of the Weatherby rifles have a long freebore. I'm not sure if the effect of that "extra" freebore would be positive or negative, but it may be something to think about, because there ARE Weatherby rifles chambered for NON-Weatherby cartridges. If You handload, this would be LESS of an issue.