I have had three scopes on my .375 H&H through the years. A fixed 4x B&L Balfor that was excellent for all around hunting. I used it in close up conditions in Quebec for a large black bear. I never felt under or over scoped. For a fixed power I like 4x on a .375 and the Leupold would be my choice.http://swfa.com/Leupold-4x33-FX-II-Riflescope-P3265.aspx
I have also had a 1.5-5x20 Leupold on it and it is also an excellent choice. It remains a backup for that gun. That particular model of Leupold has a sterling reputation among big game hunters. The glass is good, the scope compact and the eye relief is generous. It remains at the ready in Talley Q.D.'s. This model is a bit over your budget but would still be my choice if you can stretch a bit on money. http://swfa.com/Leupold-15-5x20-VX-3-Riflescope-P12632.aspx
My latest is a 1.1-4x24 Kahles. It is way over your budget. But, similar scopes in the 4200 Bushnell, and the Nikon mentioned, should be similar in overall objectives with a big game scope. Not to huge, good optics and plenty of eye relief.
The Bushnell is over your budget. The package looks to be outstanding, though. An illuminated 4A on a 1.25-4x24 is great. I also have a 1.1-4x24 Kahles with an illuminated C-dot on 45-70. It is great in low light and minute of big game for my purposes. The 4200 is a bit shorter on eye relief than the others, but still usable. http://swfa.com/Bushnell-125-4x24-Elite-4200-Rifle-Scope-P11605.aspx
I am mentioning the Nikon that was mentioned last. It is under $300, has very good glass and is a small scope. The eye relief is generous, and from my Monarch experiences, should be reliable.
For you budget, the Leo 4x and the Nikon are solid picks. If you can stretch I would easily pick the VX-3. The Bushnell is just interesting and the lit 4A is great. The only fly in the ointment is the eye relief.
On eye relief, it's just not shooting from the bench with perfect alignment that counts. For these purposes less will work. Where having plenty of it helps is when shouldering the gun quickly. That large forgiving shaft of light is very easy to pick up. That is one thing I have always liked about the Leupolds I have and have had.