I had a 742 in .308 in the late 80's...before there were so many "premium" factory loads on the market. The gun functioned flawlessly with any factory ammo....was'nt a tack driver, but plenty good enough for hunting. The only time this gun malfunctioned was with handloads. I found that I had to run my brass through the full length die twice to bring them down enough to seat properly in the chamber, and still would occasionally have a misfire. When you buy these guns new, they come with a chamber brush, for a good reason.....the chamber (at least on my gun) is very tight. I only got rid of it because I wanted to hunt with a certain bullet that was not offered in a factory load then, and it was'nt reliable enough for me with the handloads.I honestly wish I had it back, with all of the factory offerings now avaliable. I put more than 1000 rounds through that gun in less than a year, and aside from the handload issue....never had a problem with it.
On another note... I have a friend that has one (can't remember if its a 742 or a 7400) and has killed more deer with it than should be allowed, he would'nt think of using anything else.
All that being said.....yes, the BAR is probably better made, a little more accurate, and is certainly better looking, but they cost more. I've never owned one, but have had the privledge of using them in 7 mag and 243....both felt better to me than my old Remmy.
In the end... it all depends on if you want to stick to your $500 price. It'll be hard to find a Browning for under that.
Also, since you metioned "pump action" .... I do not have first hand knowledge on the 762/7600 Rem, but have heard that they are a little more accurate on average and don't have the feeding problems that i had with handloads because the pump action has more force to seat the cartridge than the recoil spring on the auto-loaders.
Hope my experience helped a little