Hello again,
Yes, having your own DNS server is the best solution.
PiHole and AdGuard are both viable choice.
Personally I use Adguard with a DNS rewrite to
*.tipi.lan.
I prefer keeping the default DHCP service on my router.*
I also recommend using another domain than fritz.box to not mess with your router.
Something like tipi.lan suits better but you can use whatever pleases you as long as you are managing your DNS.
* NB : managing your own DHCP is not mandatory because you can still edit the DNS settings separately on your devices.
You will have a better control over it but if your Adguard or PiHole encounter a problem it will cause more troubles.
Anyway, it’s a matter of choice, I suggest trying different things and decide what suits you better.