Multiplying packed 32-bit integers by a 32-bit float with AVX2

Question

I'm trying to perform a float + integer multiplication without losing too much precision. In particular, my float, mean, is in [0, 1) and the integer, value, is any uint32.

Since float32 cannot express all uint32 values, my approach was to convert value to uint64 in two 256-bit registers, convert both to float64, multiply, then, convert back to uint32. So far, I have something like (using go-asm but hopefully still gets the idea across...)

    // first, convert mean to float64
    VCVTPS2PD(mean, Y3)

    // convert deltas to uint64 in two registers
    VEXTRACTI128(U8(0), value, Y1)
    VEXTRACTI128(U8(1), value, Y2)
    VPMOVZXDQ(Y1, Y1)
    VPMOVZXDQ(Y2, Y2)

    // convert to float64
    VCVTUDQ2PD(Y1, Y1)
    VCVTUDQ2PD(Y2, Y2)

    // multiply by mean
    VMULPD(Y1, Y3, Y1)
    VMULPD(Y2, Y3, Y2)

    // convert back to int
    VCVTPD2UDQ(Y1, Y1)
    VCVTPD2UDQ(Y2, Y2)

but then I can't seem to figure out how to blend the two registers back into one register. In other words, I can't seem to find the opposite of the extract operation.

Is there a better approach to this that I'm not considering as well?

EDIT: Some clarifications that came up in the comments,

1. Each multiplication (mean, value) is a distinct pair, not a broadcasted scalar multiplication.
2. The mean is calculated from some other float32's which are also in the range [0,1). The comments suggested using a fixed point for computation, this is a viable option if it's helpful. To be concrete, the mean is a probability and the operation I'd like is to bias the value based on a mean of probabilities, for which the calculation is beyond this question's scope.
3. I have AVX2 and FMA instructions but not AVX512.
4. Round to nearest is preferred as this operation will be done repeatedly so I don't want to introduce bias, but if truncation is substantially faster I am interested in the tradeoff. This code is used in an estimation algorithm so slight bias is not catastrophic as long as it's characterized.