Improper installation of electric radiant heating mat

I recently had my bathroom renovated and heated flooring was part of the agreement, though it was not included in the original estimate. On the final day of work I spoke to my contractor who was just leaving the building and he informed me that the work was mostly finished and he would be invoicing me the following morning and hoping for immediate payment. 80% of the job has been paid immediately upon his invoicing up until this point. When I went into my suite I found that the thermostat for the floor heating mat system had finally been installed (he had told me it was unavailable for the last 3 weeks since the tiling was completed), so I tested it out to find that it was not installed in the right place (ie, right next to the doorway instead of centered where most of the foot traffic is). I emailed him about this and he said that he is very experienced in installing the product and chooses to install them "as far away from the toilet as possible". The instructions state that 6" from the toilet wax ring is the only clearance it needs. I emailed a rep for the product who confirmed that having put the mat 2' away from the toilet ring and 1' away from the base of the front of the toilet would leave excessive cold spots and that the recommendations listed in the instructions for installation are based on standard electrical and building codes, and to give heated coverage as close to 100% of the most used areas of the floor. My contractor is demanding the last payment, has not apologized or offered a solution, has not admitted that there is anything wrong with where he has placed it, and is trying to guilt me that he is having to take out a loan to pay his workers, and that he has made almost no money on the project as he was trying to make it as affordable as possible for me (even though I received two other very comparable quotes in terms of labour costs). I have received one ball park quote that to re-do the floor will cost more than he claims I owe him. My question is: Am I obligated to pay the last invoice? Do I need to give him the option in writing to re-do the job at his expense if he wants final payment? I really don't trust this person anymore and am hoping I can hire someone else to fix it. If it wound up costing less than the last invoice I'm fine paying him that difference.