There are two alternatives sounds to me like. Either a "Wrong Wrench" or a "Wrong nut", not to be too flippant about it.
Savage went to a larger barrel shank when the WSM calibers came out. Measure the OD of the barrel shank where the nut threads on the barrel. The "Standard" barrel measures 1.055" x 20 tpi and the "Large" barrel measures 1.120" x 20 tpi, but I think the OD of the nut was the same. I've never encountered a nut that my wrench wouldn't fit, large shank or standard shank.
I used a pipe wrench to get a nut off the first time I did a barrel swap because I didn't have a wrench. Was careful to hook the teeth into the square notch. It worked OK. Very slight damage that was easy to clean up and reblue even using a brass shim between the lower jaw and nut.
for lots of good information on Savage.
Call Midway, tell them your problem and they will fix it up. Or...
You can order the correct wrench from Sharp Shooters, return the wrong wrench to Midway and continue your project. Be sure to measure the OD of the nut where the wrench goes so you can compare it to another factory barrel nut.
I've done many savage barrel swaps without any problems going way back to the mid 60's. I used various methods for the older round notched nuts. When all the hoohaa with Savage started I purchased my barrel wrench from Sharp shooters Supply and continue to buy from him. He's good people.
I can't understand the nut diameter difference. Wheeler Eng usually produces excellent products.
The other alternative is an incorrect dia nut installed either by the factory or some one else. In that case I would buy the correct size nut from Sharp Shooters when you get things sorted out.
Hope this helps.