Shouldn't be a problem to get a shower in the business class lounge. I've never seen all four showers (two in the men's area [No 1+2], two in the ladies' [No 3+4], but they don't make a difference, they once sent me to the ladies' showers despite me being anything but a lady) to be in use at the same time. Just hand in your boarding pass at the reception in exchange for the shower key.

However, be aware that the showers (at least the one I'd been sent to) have a bit of a bad interior layout with a really strange shower head, and you are very likely to get your clothes wet if you use the coat hanger - this may be better in the Arrival Lounge, but I cannot really tell as I've never been to the Arrival Lounge.