I would settle for a quality .30-06, with controlled round feeding, wearing a quality scope, (Schmidt & bender or Swarovski variable.) Anything from a 7mm to a .338 will work, but as I recon you should stick to medium and smaller antilope with the light rifle (.30-06), and use it on eland and kudu only in emergency, I went lower than .338.
You have a .375, which IMO should be used on eland and kudu size animals. However, whatever caliber you decide on, learn to handle it well, and use only premium quality bullets.