This isn't silly at all, it shows you care about protecting your family from a very real threat.
Laws vary a lot between jurisdictions in the US, so the details will vary depending on which city, county, and state the death occurs. Any suicide will be treated as a suspicious death, so the police and the medical examiner or coroner's office will open independent investigations.
The police will determine wheter a crime has been committed, and they will notify the next of kin of the death. They will probably ask the next of kin for identification, and most likely just a driver's license or state ID card would be sufficient for that. This is where location makes a big difference. If you're in a liberal state or jurisdiction the police might
not be allowed to ask about immigration status, and they would
not cooperate with agencies like ICE on immigration enforcement except in limited circumstances. The
California Values Act is an example. So if you're in one of these so-called sanctuary jurisdictions, you wouldn't have to worry much about the death investigations causing family members to get swept up by ICE. However outside those jurisdictions, especially anti-immigration states in the South, it is possible the police could contact ICE if they suspect immigration law violations.
The medical examiner or coroner will investigate the cause of death, and they may need to communicate with your family directly or through the police. I think it's unlikely the medical examiner or coroner would be interested in immigration enforcement. However from your family's perspective they are government agents working closely with police, and it probably would be very stressful for your family members to interact with them, so that's something to take into consideration if you decide to CTB.