Oxi Clean works great on tile and grout, you can use 4 oz in a bucket with a mop or add 2 oz to the steam cleaner reservoir and fill with warm water. Really dirty grout will require you to get on your hands and knees with a toothbrush or scouring sponge.

Cleaning With Oxi Clean

"or add 2 oz to the steam cleaner reservoir and fill with warm water"

As far as I know, you cannot add anything to the water reservoir of a steam cleaner.

If you don't want to spend much money and you have limited storage space, then maybe a steam cleaner is not the answer for you.

You have three different types of tile - marble, granite, and (I assume) ceramic tile. Is there grout around the marble and granite, or just around the ceramic tiles?

I have not used a steamer on marble and granite in a bathroom. A steamer might melt any sealer that is on the marble or granite. What products have you tried thus far? You might try the Stone Tech line: Stone Tech

Afraid all three tiles have grout (not to mention husband's shower floor of glass mosaic.)

My experience with my Sargent Steam is that it works on the soap scum and dirt, but it doesn't do much to mineral deposits. Also, we, like you, have a very tight space to work in in our bathrooms in an old San Francisco house. It's a struggle to get the wheeled console into a position that makes it easy to clean the shower and around the toilet. Our water is not too hard, and I've found that a Krud Kutter solution works great. If your grout is not sealed, get it clean and then seal the grout with an oil-based sealer made for wet-location use.

I have had a Shark steam cleaner for years. I bought while listening to QVC rave about how I couldn't live without it so I bought it. Turns out I could live without it but that's another story. It does an ok job on most things but I would NEVER add oxi-clean to the reservoir. I agree with graywings that most steam cleaners tell you not to add anything. Maybe you could wet the tiles w/oxi clean & then steam it but that sounds like too much work to me. I use mine only occasionally to steam my mattress. I cannot remember when I have used it for anything else in several years. Save your money & take one of the suggestions above.