I've been in the flooring game selling carpets timber vinyl and ceramic tiles for 35 years, you are not the only one with this problem. The best solution is to remove the tiles but this is a big job but gives you 100% job but you probably know this. Bamboo flooring is good but please do your research on this because they are made two different ways Cold press and Hot press and the prices very from $30 - $60 also you will need to ask for the VOC certificate for things like formaldehyde. A lot of the cheaper suppliers do not have there's tested. The other thing that the industry is having but they want tell you is the bamboo is expanding and contracting a lot more in the Australian climate. Sorry to put a damper on bamboo. There is a newer floating floor on the market it has been around for about 7 years it is vinyl 4.5mm to 6mm thick its is good for going over tiles as it does not create to-much of a step and other issues in doors and kitchen

one problem with laying over the top of an existing floor is that it will create small steps at the doorways and cover up any potential problems that will be even more difficult and expensive to remove after the new floor is installed...i acknowledge that removing the old tile or floating timber floor will be dirty hard work but it shouldn't be too difficult but i suggest that some expert advice will help identify the option that will achieve the best results and removing the skirting boards and replace after the new timber floor is installed for an even better result

What did you decide to do?

Agree with oklouise, check out Dulux renovations range, they have a floor paint, they have a how to video , I have used the one for Ben tops and cupboards and it is great, it’s available at Bunnings.