I think it's fine. Anyone who would be offended by being wished a Merry Christmas has a screw loose in my opinion. America is a predominantly Christian country - and the shopping season is busy at Christmas because this is when Christians shop. If you wish a non-Christian a Merry Christmas, and are corrected, then wish them a Happy Holiday Season.
The end of Ramadan is the busiest shopping season in Sarajevo - with October and November being the busiest months overall. During that time, everything is Ramadan - Ramadan Sale! Etc. But in December, everything is Christmas. Even stores selling traditional, Islamic clothing will have Christmas sales - because Christians have Muslim friends, normally. So it depends on the season.
There's a Ramadan season, a Christmas season, and whatever else. The signs change, the greetings change, but they're never just merged into one benign expression. I prefer that.
I prefer going to a Christian-owned shop in November and getting wished a happy Ramadan, just as I prefer serving Christian customers in December and wishing them a merry Christmas. Saying Happy Holidays just annoys everyone, here it would make non-Muslims all assume it means Happy Ramadan anyways.
Besides, we're not celebrating "Holidays". I don't buy presents for the long weekend in the first week of July. I don't buy presents when snow closes the businesses. I buy presents for Ramadan, and for Christmas. This is what we're celebrating, so this is what we should wish for each other.
I think "Happy Holidays" also requires a fairly secular people, in beliefs and not just in lifestyle. Ramadan and Christmas both, in Southern and Eastern Europe, are still far more religious than they are commercial. If you're a Christian in Sarajevo and don't want to celebrate the religious aspect of Christmas, there's really nothing you can do. There's no big Santa Claus mission or anything like this. It's just simply presents to celebrate a religious holiday.
So I think having such a strong religious aspect of these holy days makes it impossible to just make them holidays.