Should it be possible for a single malfunctioning IRU to cause a noticeable shift in the FMC position (thus a mapshift on the EHSI) if all other sensors are working fine? The short answer is no it should not, but it may.
On non-GPS equipped aircraft, FMC position is biased approximately 80:20 towards radio position and the position determined by the L IRS. So if the L IRS was drifting it could influence the FMC position.
On GPS equipped aircraft, FMC position is calculated from GPS and ADIRU inputs. The FMC uses its onside GPS (or the offside GPS if the onside GPS data is invalid). No radio updating occurs, unless both GPS signals are lost or data is invalid.
BTW if you suspect the integrity of an IRS, you should select the POS REF 2/3 page after flight when you have come to rest and see what the residual groundspeed is of the IRS’s. The limits are laid down in the AMM, but as a rough guide any more than 6kts per hour of flight (after a full align) is unacceptable. Below is a photo of one I snagged last year after a 2hr airtest.
S&L
(Edited for typos and to add photo)