they would have to get permission from Big west to use these designs or have the them approve any deviations from them?
Except that, as far as Battletech is concerned, those original Big West designs have had a retgone done on them, and the new artwork that we're getting is "what they looked like all along." It shows a good-faith attempt to redress the situation by using new artwork that is not copyrighted by Big West, while still having art that matches the stats - and there will always be similarities in "warrior robots" that you just can't get past. When you want a warrior robot with a missile launcher on it and some big guns for arms, well...you get it.
Scenes a faire is what Leonard French called it a while back. Things that are going to be visually similar just because it's the basic nature of the thing. A depiction of a battleship, for example, is going to have turrets with guns on it, a general slim 'football shape' to its lines, and various towers for control of the ship, radios, radars, exhausts, and so on. A tank, meanwhile, will generally have one big gun on it, maybe a smaller one next to it, a box shape overall, tracks, and look very heavily armored. A humanoid warrior robot is going to have two legs, two arms, a head, a body, and guns all over it in various logical places (usually) whether they're cannons or lasers or missiles. Similarity in design isn't always copyright infringement, if there's obvious function dictating form.