Is there a problem - except with the filesize - when using bmp files for websites?
Lack of support? I wouldn't trust many browsers to handle them properly.
With browsers post-2002 or with Internet Explorer, there is no problem, they will display it. IE supported it and the others (you can find the code for Mozilla/Firefox and for Chrome/WebKit) followed suit to stay compatible. The only issue is indeed size, which means:
If you need lossless compression, use PNG: it's widely supported, and the few browsers that do not support it completely, like IE 6, do support all the features BMP has too -- they just don't support PNG transparency but BMP doesn't have transparency either.