This is not how most people calculate the yield of a floater.
The way most people calculate the yield of a floater is:
1 for each remaining unset coupon, project the values of the index that will be used (such as 3Mo LIBOR, daily SOFR, SONIA, ESTR, etc - see Forecast 3m LIBOR USD. Budget purpose for example); and project the coupons. For example, if a floater has a coupon that in 1 year is reset from 3 months LIBOR plus 100 basis points ("quoted margin"), and if the projection curve predicts that 3 months LIBOR will be 30 basis points when this coupon is reset, then you project that the coupon will be 30+100 = 130 basis points.
2 Just as you would for a fixed-coupon instrument, solve for the internal rate of return (IRR) of the cash flows where you pay the dirty price on setlement date and receiver the set coupon(s), the unset coupons projected in step 1, and principal repayments.
Various spreads that make sense for fixed-coupon bonds (Z-spread, OAS, etc) work just as well for floaters, discounting the future cash flows by your bond funding cost. You can also calculate the discount margin (DM), which is similar to these spreads.
(Sometimes people calculate "flat yield" and "flat discount margin" by taking the current value of the index and assuming that it will remain constant, so you don't need to project it from the curve in step 1. This is the default behavior of the Bloomberg Terminal - a setting which you may want to change.)
The yield of a floater changes every time the projection curve changes, so, unlike fixed-coupon bonds, floaters are almost never quoted on yield. Floaters are usually quoted on price or on DM. Backing out price from DM (the inverse of the price to DM calculation) is depends somewhat on the projection curve assumptions.
If you want to reprice the bond under various risk scenarios, then you should not assume that your discount curve (funding) and the projection curve (used to predict unset coupons) are the same. But the two curves will be correlated. In particular, for a dv01 scenario, you may want to perturb both discount and projection curves by the same 1 basis point up and down.
If the DM moves (e.g. because the bond issuer's credit changes), then you back out the new bond price (the inverse of the price to DM calculation), and use the new price to calculate the new yield, Z-spread, and other spreads, which will all change by about the same amount as the perturbed DM.
As you see, there's no synthetic fixed-coupon bond in this picture. You could make one up, but I don't see how that would benefit anyone in understanding the value of the floater.