How do traveller academics get reimbursed for accommodation shared with families?
Usually, you will explain the specifics of your travel and provide some documentation about


the actual expenses for travelling with your family
the expenses you would have had, had you travelled alone


You should then be able to make a case for getting reimbursed for the (lower) expenses listed in item #2, even though all your bills show the (higher) expenses from item #1.

If there was anything that cost the same, or even less, for the whole family compared to what you would have paid alone, make sure to provide good proof of the hypothetical 1-person-price and you might even get reimbursed some of what was used by the entire family :)