It will punch you pretty good, "too stout" is up to you. However you could get the gun out of the custom shop and have them make it in one of the other cartridges (7mm mag, 308, etc). You could also have them install an accubrake. Any weatherby dealer that understands the custom shop should be able to write up the order. Give them 6-9 months if your going to do a custom setup.

I wouldn't want to shoot an elk with 120gr .257 bullet, but its legal.

The stock design will help, but I wouldn't touch the .300. I had a full Mark V Deluxe in that was as much as I care to handle.
Without running numbers, I'm betting that thing will be a lot like this little pump with 2 oz loads. A few rides on it will give you a headache. I only shoot it b/c I am nuts for turkeys and spend tons of time chasing them.

Its find blurry pics in my photobucket day, btw.

The 300 WBY with a muzzle break is very manageable, but the noise is ATROCIOUS. You may lose friends over it (the first time you fire it next to them) and if you go on a guided hunt you will get a glare from the guide that could melt steel. Each time you fire it without hearing protection you WILL DO PERMANENT DAMAGE to your hearing (as with any big rifle, but much more so with this one). Personally, if it needs a break I don't want it. But I do understand how hard it is to find a good left handed rifle that you like.

On the other hand, 120 grains is a bit on the light side for bigger game but there are people that do it. If I were to go that route I'd be carrying a premium bullet for sure, but I personally would explore other options. Just an opinion though.