Minimum
Why 253 and not 250 since feerate-per-kw should be feerate-per-kvb / 4 and the minimum relay fee rate is 1000 satoshi/kvb (see [MinimumRelayFeeRate])?
Because Bitcoin Core uses neither the actual tx size in bytes nor the tx weight to check fees, but a "virtual size" which is (3 + weight) / 4. So we want: fee 1000 * virtual size feerate-per-kw * weight 1000 * (3 + weight / 4) feerate-per-kw 250 + 3000 / (4 * weight)
With a conservative minimum weight of 400, assuming the result of the division may be rounded up and using strict inequality to err on the side of safety, we get: feerate-per-kw 252 hence feerate-per-kw >= 253