Small ring Mausers
As DAN BELISLE stated, it is possible, and he has given you some good advice on which models.
I would tend to disagree about the m/96 Swedish Mauser, though. Many of these have been made in 30-06, 9.3x62, and 8x57 JS. These calibres are in the same pressure class as the .280 Remington.
I would also be a bit wary of the original WWI German Karbine 98 small ring models that were made during the later years of the War. The better option is use the standard 98 Mauser large ring actions. The weight difference is only about 4 or 5 ounces, and many aftermarket options such as stocks, triggers, etc., are available.
Trade-ex has lots of listings of the 96 Mausers in the Calibres I mentioned above (except the .280,) and they have the Husqvarna 1600 series rifles available in the $400-$500 price range. The 1600 series IS a small ring Mauser, with the third locking lug, with a sporting stock, trigger system, safety, and are usually drilled and tapped for scope mountings.
Since you are saying you have a 6.5x55 rifle now, can we assume you already have a m/96 Mauser, and it might not be so simple as just doing the barrel. The magazine might have to be lengthened and feed lips may also have to be modified. This is going to be an added expense, along with the cost of the barrel, fitting, chambering and other gunsmith work.
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