You might find this thread interesting: Colleges ranked by SAT Score: What do you find interesting?. In looking at these schools’ average SAT scores from 2015 (i.e. pre-test optional almost everywhere), these are some of the schools with very similar SAT scores (within +/- 10 points).
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Butler (IN): About 4500 undergrads: Actually has the exact same average SAT as Miami of Ohio, U. of Denver, and College of Wooster and within 10 of Fordham, Skidmore, Furman. As it’s in the state capital, there are also lots of political opportunities here.
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Christopher Newport (VA): About 4500 undergrads. Within 10 of DePaul, U. of Cincinnati, Lake Forest, Bradley, Allegheny, SUNY Buffalo, U. of Iowa, Iowa State, U. of Nebraska, U. of Kentucky, etc.
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Mercer (GA): About 4900 undergrads. Within 10 points of New Jersey Institute of Technology, Clarkson, Penn State, Loyola Marymount, St. Lawrence, U. of Delaware, Texas A&M, U. of Vermont, etc.
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Truman State (MO): About 3900 undergrads. Has the exact same score as Elon, Cal Poly-San Luis Obispo, CUNY-Baruch, Wofford, Loyola Chicago, U. of Colorado, Texas Christian, and U. of Alabama - Huntsville, among others.
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U. of Alabama - Huntsville: About 7600 undergrads. See Truman State.
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U. of Denver (CO): Already mentioned; about 5900 undergrads: See the schools mentioned for Butler
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U. of North Carolina - Asheville: About 3200 undergrads. See the schools listed for Mercer.
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U. of Tampa (FL): About 9900 undergrads. Within 10 points of UNC-Charlotte, SUNY-Oneonta, Towson, U. of Illinois - Springfield, U. of Montana, U. of Maine
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U. of the Pacific (CA): About 3300 undergrads. Within 10 points of UC-Irvine, Indiana University, Syracuse, Fairfield, Loyola Maryland, U. of Vermont, Texas A&M, etc.
On College Navigator, at least, it indicates no religious affiliation for Mercer: College Navigator - Mercer University. Its Mission & Vision indicates that it was founded by Baptists, but indicates that it is now independent.