there was actually no need for you to have spent money to apply for the tourist visa, as it only grants the same 60 day stay (+30 day extension from immigration offices in Thailand) as visa exempt entries for UK passport holders. It serves no purpose to normal tourists since they changed visa exempt entry from 30 to 60 days in mid-July.
The form is not called TM90. It is actually TM47. The name is the government form number. It doesn't have anything to do with it being due every 90 days.
If you entered visa exempt or via tourist visa, you can only stay 90 days anyway even with an extension, so you must have some other way you entered that is allowing you to stay longer than 90 days, therefore requiring the 90 day report. If you are planning to illegally overstay for 5 days, then the 90 day report is the least of your worries.
Make sure you know the difference between a TM30 and a op day report. TM30 is required within 1 say of arriving at any address in Thailand to stay overnight, and is done by the homeowner to advise the government that a foreigner is staying there. Hotels do this each day for guests that check in that day. A 90 day report is a self notification that you must inform immigration where you are staying every 90 days you remain in the country. If you exit, the 90 days resets to count from your next entry date. You need a TM30 (initial report of where you are) to file a 90 day report.
Change your travel agent if they aren't professional enough to do a simple Google search or even look up the visa policy of Thailand on Wikipedia, or even better, on the website of the Thai embassy in Australia, especially after you inform them they are mistaken. Also shows how seldomly this agent must deal with bookings to Thailand if they haven't kept up with Thailand's visa exempt policy change that has been in effect for several months already. Maybe find an agent more familiar with Thailand to help with your booking? Otherwise they are likely pulling travel advice from the same general googled sources as you could do yourself and therefore add no value to using them.