Sure, a mechanical relay will be OK for occasional use if you slow down the control system response enough and don't need to hold the temperature closely, but its going to wear out fairly quickly due to contact burn if its breaking a large proportion of its rated current several times a minute. If you can afford an appropriately rated SSR, its an easy drop-in replacement for the existing relay. As you are driving it from your own code in the MCU, it would be fairly easy to add isolated load current sensing to detect any failures and add an audible alarm for it. Whether or not you need to add a thermal trip depends on the consequences of severe over-temperature. However as the most problematic time to have uncontrolled full power would be while still melting the PLA out of the mould due to the risk of igniting the PLA in the drip tray, you might want to simply keep the existing relay in series with the SSR so the controller can cut the power if it detects a SSR failure or other fault. The mechanical relay woud be constanly on in normal operation. Redundant temperature sensors would be a good idea - if there is too much of a difference between them, treat it as a fault and shut down.
If you aren't going to use the burnout kiln regularly, you may wish to build up the controller as a detachable unit with a socket for controlled mains out, and plug in connectors for the temperature sensors so you can use it for other heat control applications. You may even decide to add the capability to load settings and temperature profiles from SD card or via USB to make it more versatile.