there is an expectation that if you arrange your own accommodation that it is reasonable quality and a reasonable price but at some destinations at some times of the year even a mid range (reasonable to most people) can't be had for a reasonable price
BUT there are no limits on room rates listed in EU261 and whilst an airline might have a policy it cannot be an absolute limit. €80 seems very low to me. The BA 'policy' is £200 for example and that still causes problems for some people.
There is a claim form / email on the tap website. You must given them a change to process your claim - you can't just jump down the legal route. Yes a claims handler could do it for you but since you'll have to provide them with all the info anyway you might as well do it your self.
Refusing to take the flight they offer does not automatically preclude them paying delay compensation it can limit your chances of getting it. Their obligation is to get you to your destination ASAP and if you chose to get a later flight there is an argument to be made that you extended the delay yourself and so no compensation is due. Though in your case given you were going to be delayed anyway thus meeting the EU 261 criteria they are still likely to pay but may initially push back.
Keep your email simple - stick to the facts ' this happened on X day and I claim compensation under EU261 and I spent Y on a hotel you need reimbursement for etc etc