Take a look at some of the national organizations and their regulations. I'm familiar with NASA and their CCR has details about cage minimums and requirements that are easy to follow. You'll also need to have an understanding of what class the car will likely fit into as there are some class-specific rules to pay attention to. Considering your build description, you'll likely fall into the "unlimited" or "open" type classes so you should have lots of freedom for extra chassis stiffening and cage attachment points beyond the typical 6+2.
NASA CCR:
https://nasa-assets.s3.amazonaws.com...282/2023.1.pdf
SCCA CCR:
https://cdn.connectsites.net/user_fi...pdf?1704409574
It's also worth mentioning that autocross and time trials are not the same as actual wheel-to-wheel racing despite the blanket "racing" term people like to use. You'll find significantly different safety requirements for TT vs W2W depending on the organization. For example, NASA TT you can run a street car in stock format right off the street. However for real W2W racing, you'll need a 6+2 cage with specific design and tube thickness based on the car, fixed seat within expiration date, harness within expiration date, fire system, electrical cut off, and many other things that meet the rules of your organization's CCR.