I am having trouble finding a new mattress to fit a bassinet I got second hand. I have read on forums that you can take the existing mattress to Clark Rubber and they can match the size, thickness and type of foam. Is this safe to do for a newborn?

Red Nose recommends that a mattress is firm (ie meets voluntary standards for firmness) clean, well-fitting & flat.
https://rednose.org.au/article/what-is-a-safe-mattress - Cot mattress.

Bassinet mattress - What to look for in a bassinet   https://rednose.org.au/article/bassinets
Mattress no thicker than 75mm, & height of side of bassinet at least 300mm high from mattress top.
Red Nose does not have a recommendation for fabric type.

How old is the bassinet?
Please see information about second hand nursery furniture
https://rednose.org.au/article/second-hand-nursery-furniture

There are no current standards for bassinet up to June 2024. These standard will become Mandatory in Jan 2026.
Have a look also, at Bassinet information from Product Safety Australia.
https://www.productsafety.gov.au/consumers/keep-baby-safe/settle-baby-to-sleep-safely/bassinets-and-cradles-guide