By "all around DNS solution" I assume you mean you are running unbound on the Pi as the local DNS recursive resolver. If so, this local resolver talks directly to the root servers listed in "/var/lib/unbound/root.hints," and these in turn point the resolver the lower domain servers as appropriate. The PiHole talks to the local resolver, the local resolver goes to the appropriate domain name servers and gets the DNS, then the PiHole gives the DNS back to the client.
I run this setup on a Pi, but I don't have the same router as you and I don't use the PiHole as my DHCP. However, I don't think either of those two items change the behavior.
I don't believe that the local resolver talks to your router DNS in any way - it is completely bypassed in this setup as long as your network devices are all using the PiHole for DNS, which they should be if you use the PiHole DHCP.
Run "dig pi-hole.net" and see if the resolver is talking to the root servers. In the case shown below, it is.
pi@Pi-3B:~ $ dig pi-hole.net
; <<>> DiG 9.10.3-P4-Raspbian <<>> pi-hole.net
;; global options: +cmd
;; Got answer:
;; ->>HEADER<<- opcode: QUERY, status: NOERROR, id: 58442
;; flags: qr rd ra; QUERY: 1, ANSWER: 1, AUTHORITY: 6, ADDITIONAL: 13
;; OPT PSEUDOSECTION:
; EDNS: version: 0, flags:; udp: 1472
;; QUESTION SECTION:
;pi-hole.net. IN A
;; ANSWER SECTION:
pi-hole.net. 86400 IN A 206.189.252.21
;; AUTHORITY SECTION:
pi-hole.net. 86400 IN NS ns6.pi-hole.net.
pi-hole.net. 86400 IN NS ns3.pi-hole.net.
pi-hole.net. 86400 IN NS ns2.pi-hole.net.
pi-hole.net. 86400 IN NS ns4.pi-hole.net.
pi-hole.net. 86400 IN NS ns1.pi-hole.net.
pi-hole.net. 86400 IN NS ns5.pi-hole.net.
;; ADDITIONAL SECTION:
ns1.pi-hole.net. 13045 IN A 208.94.148.4
ns2.pi-hole.net. 13045 IN A 208.80.124.4
ns3.pi-hole.net. 13045 IN A 208.80.126.4
ns4.pi-hole.net. 13045 IN A 208.80.125.4
ns5.pi-hole.net. 13045 IN A 208.80.127.4
ns6.pi-hole.net. 13045 IN A 208.94.149.4
ns1.pi-hole.net. 13045 IN AAAA 2600:1800:10::1
ns2.pi-hole.net. 13045 IN AAAA 2600:1801:11::1
ns3.pi-hole.net. 13045 IN AAAA 2600:1802:12::1
ns4.pi-hole.net. 13045 IN AAAA 2600:1801:13::1
ns5.pi-hole.net. 13045 IN AAAA 2600:1802:14::1
ns6.pi-hole.net. 13045 IN AAAA 2600:1800:15::1
;; Query time: 32 msec
;; SERVER: 127.0.0.1#53(127.0.0.1)
;; WHEN: Tue Jul 10 10:26:40 CDT 2018
;; MSG SIZE rcvd: 428
----------------------------
If you run the same command again with an external DNS server (bypassing your local recursive resolver), the authority section is not there.
pi@Pi-3B:~ $ dig pi-hole.net @1.1.1.1
; <<>> DiG 9.10.3-P4-Raspbian <<>> pi-hole.net @1.1.1.1
;; global options: +cmd
;; Got answer:
;; ->>HEADER<<- opcode: QUERY, status: NOERROR, id: 21777
;; flags: qr rd ra; QUERY: 1, ANSWER: 1, AUTHORITY: 0, ADDITIONAL: 1
;; OPT PSEUDOSECTION:
; EDNS: version: 0, flags:; udp: 1452
;; QUESTION SECTION:
;pi-hole.net. IN A
;; ANSWER SECTION:
pi-hole.net. 8534 IN A 206.189.252.21
;; Query time: 24 msec
;; SERVER: 1.1.1.1#53(1.1.1.1)
;; WHEN: Tue Jul 10 10:44:11 CDT 2018
;; MSG SIZE rcvd: 56