Chris is self-employed, isn't he ? I guess that pretty much rules out his expenses being charged to his business.
Chris' finances are indeed shrouded in mystery. There have been reports in the past that his business ventures were not doing so well and he might even have had some financial trouble, but no concrete evidence has ever shown up. Madeleine, on the other hand, is a millionaire in her own right and might be footing part of the bill. There is always speculation that she may also get some money from her father although, if she does, the money probably comes from Carl Gustaf's private funds and not from the State appanage.
The appanage itself is a block grant paid to the Royal Court of Sweden and currently divided up between the King's Household, the Queen Consort's Household, the Crown Princess' Household, and the Royal Mews. A separate block grant is destined to the maintenance and operation of the Royal Palaces. Madeleine and CP do not get any money directly from the Swedish taxpayers, although they may be reimbursed from appanage money when they perform official duties on behalf of the King. As mentioned before, Madeleine's wedding and the renovation of her apartment in the Royal Mews were partially paid for with State funds.
Royal Finances - Sveriges Kungahus