The service entrance wiring is totally unfused
Service entrance wiring runs from your weatherhead or underground splice point, through the meter, and onward to the main breaker.
If the wires shorted together, no breaker anywhere will protect it.
This wire is also not grounded.
As such, the service entrance wires are not allowed to take "the grand tour of the house" as it were. Code requires they be absolutely as short as possible - typically with the main panel being on the other side of the wall from the meter pan.
So "hard no" on the idea of sending these wires totally unfused through the house.
NEC 2020 changes everything, though.
As the 2023 NEC edition has come out, many states are getting serious about adopting NEC 2020 and most have.
NEC 2020 requires an outside disconnect - the concept is to give firemen an easy way to cut power. But the effect is that the reasonable way to comply with the rule is a Meter-Main. This is a "meter pan" with a main breaker added.
Even if you're not on NEC 2020, the effect of the rule has been to change the economies of scale of meter-mains so they're pretty cheap now. So go ahead and use one.
Wow! That nicely solves the "unfused wiring" problem above! This main feeder is allowed to take the grand tour of the house if it wants.
It is not service wire since it's past the main breaker. It is feeder, so it needs 4 wires. But it sounds like you have that.
Individual wires need conduit. If you want cable, use cable.
Individual THHN, XHHW or USE/RHW wires need to be inside a conduit. If you want to cross the house without use of a conduit, use a cable that is approved for direct use indoors per NEC Chapter 3.
The first one that comes to mind is SE-R.