The Le Anfore lounge does have showers, in the back of the lounge on the right-hand side.
As an EXP, you should be able to use the BA lounge. I've even had some success using the BA lounge when not flying oneworld...the BA lounge has the only legal smoking lounge in the non-Schengen departures area of FCO. Most Italians (including the lounge staff) can sympathize with me on this point.
Neither lounge is anything special, and both are better than the Alitalia lounge. Of the two, I prefer the BA lounge, as it is less crowded during the peak am departure hours.