In my experience, overgraded, yes; trimmed, no. Most of the time I wouldn't bid more than 2.5 grades less than the BVG grade to cover risk and PSA grading costs, in this case PSA 4.5ish money. Most BVG cards I've cracked and subbed to PSA come in 2 grades lower, though I have had a few come back the same grade.
Beckett will grade sheet-cut cards, as long as they are the correct size or larger. I have experience with that, but not necessarily a trimmed card. Since they grade sheet-cut, I would assume they will also grade trimmed as well.
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WTB: PSA 1 - PSA 3 Centered, High Eye Appeal 1950's Mantle