So many holes in your argument/logic, @Totupthenumberspls
First you say you've been asked to go for a week, then it becomes 10 days. Which is it?
Next you say you and your partner share childcare 50:50, yet you then say it's you who does all the pick ups/drop offs
... but you also acknowledge that your partner could manage without you
Then you say your company is very 'blaze' (I think you mean 'blasé) about your contract, when very specifically - i.e. not blasé - it says:
unlimited hours, travel as required, location as deemed necessary
And go on to start about them profiting from you going on the trip. Well of course! The whole point of having employees in a business is to help the business make money!
Look. The cost of childcare is a cost to you of going to work. Just like the cost of travel. If you suddenly moved 200 miles away, because you prefer the new area/house, you wouldn't expect your employer to meet the cost of travel, would you?
You're making something out of nothing here. You have admitted that your partner can cope when you're away (for however long). So what is the issue? You're being awkward for the sake of it.
The expenses that would be justifiable to claim are those that are to do with the trip - so out-of-pocket expenses when you're there, like subsistence, client entertainment, travel to/from hotel, etc. And if the trip means you're away for a weekend, which would be the case for a 10 day trip, then you should be able to claim TOIL as well.
But to moan on about needing extra and unnecessary extra childcare is ridiculous.