Ok, I tried what you told me at home (got the same problem here in my home LAN)

1) Yes, I can access the configuration interface of the printer  via WLAN

2) Yes, I can ping the printer (over WLAN and LAN)


------- At first all the commands, tested while connected to network via WLAN --------

3a) Running lpinfo -v (while on WLAN) results in:
network lpd
network ipp
network ipps
network https
network http
network socket
network beh
direct hp
network smb
direct hpfax
network lpd://BRNF3DB2C/BINARY_P1

3b) running sudo /usr/lib/cups/backend/dnssd (while on WLAN) results in:
-> no response (I had to ctrl + C to stop the command)

3c) running /usr/lib/cups/backend/snmp (while on WLAN) results in:
network lpd://BRNF3DB2C/BINARY_P1 "Brother MFC-9840CDW" "Brother MFC-9840CDW" "MFG:Brother;CMD:PJL,PCL,PCLXL;MDL:MFC-9840CDW;CLS:PRINTER;" ""

3d) running route (while on WLAN) results in:
Kernel-IP-Routentabelle
Ziel            Router          Genmask         Flags Metric Ref    Use Iface
default         speedport.ip    0.0.0.0         UG    0      0        0 wlan0
link-local      *               255.255.0.0     U     1000   0        0 wlan0
192.168.1.0     *               255.255.255.0   U     2      0        0 wlan0

3e) running nmap on the address of the rounter (while on WLAN) results in:
Host is up (0.014s latency).
Not shown: 994 closed ports
PORT     STATE SERVICE
21/tcp   open  ftp
23/tcp   open  telnet
80/tcp   open  http
515/tcp  open  printer
631/tcp  open  ipp
9100/tcp open  jetdirect

------ Now the same while connected to the LAN via cable -------

4a) running lpinfo -v (while on LAN) results in:
network lpd
network beh
network https
network ipps
network socket
network ipp
network http
network smb
network dnssd://Brother%20MFC-9840CDW._ipp._tcp.local/
network dnssd://HP%20Color%20LaserJet%20CM2320nf%20MFP%20(8869EE)._pdl-datastream._tcp.local/
network dnssd://Brother%20MFC-9840CDW._pdl-datastream._tcp.local/
network dnssd://HP%20Color%20LaserJet%20CM2320nf%20MFP%20(8869EE)._printer._tcp.local/
network dnssd://Brother%20MFC-9840CDW._printer._tcp.local/
direct hp
direct hpfax
network socket://192.168.1.11

4b) running sudo /usr/lib/cups/backend/dnssd (while in LAN) results in: 

DEBUG: Found "Brother MFC-9840CDW._ipp._tcplocal"...
DEBUG: Found "HP Color LaserJet CM2320nf MFP (8869EE)._pdl-datastream._tcplocal"...
DEBUG: Found "Brother MFC-9840CDW._pdl-datastream._tcplocal"...
DEBUG: Found "HP Color LaserJet CM2320nf MFP (8869EE)._printer._tcplocal"...
DEBUG: Found "Brother MFC-9840CDW._printer._tcplocal"...
network dnssd://Brother%20MFC-9840CDW._ipp._tcp.local/ "Brother MFC-9840CDW" "Brother MFC-9840CDW" "MFG:Brother;MDL:MFC-9840CDW;" ""
network dnssd://HP%20Color%20LaserJet%20CM2320nf%20MFP%20(8869EE)._pdl-datastream._tcp.local/ "HP HP Color LaserJet CM2320nf MFP" "HP Color LaserJet CM2320nf MFP (8869EE)" "MFG:HP;MDL:Color LaserJet CM2320nf MFP" ""
network dnssd://Brother%20MFC-9840CDW._pdl-datastream._tcp.local/ "Brother MFC-9840CDW" "Brother MFC-9840CDW" "MFG:Brother;MDL:MFC-9840CDW;" ""
network dnssd://HP%20Color%20LaserJet%20CM2320nf%20MFP%20(8869EE)._printer._tcp.local/ "HP HP Color LaserJet CM2320nf MFP" "HP Color LaserJet CM2320nf MFP (8869EE)" "MFG:HP;MDL:Color LaserJet CM2320nf MFP" ""
network dnssd://Brother%20MFC-9840CDW._printer._tcp.local/ "Brother MFC-9840CDW" "Brother MFC-9840CDW" "MFG:Brother;MDL:MFC-9840CDW;" ""

4c) running /usr/lib/cups/backend/snmp (while in LAN) results in:
network socket://192.168.1.11 "HP Color LaserJet CM2320nf MFP" "HP Color LaserJet CM2320nf MFP" "MFG:Hewlett-Packard;CMD:PJL,PML,PCLXL,POSTSCRIPT,PCL;MDL:HP Color LaserJet CM2320nf MFP;CLS:PRINTER;DES:Hewlett-Packard Color LaserJet CM2320nf MFP;MEM:MEM=111MB;COMMENT:RES=600x8;" "HP Color LaserJet CM2320nf MFP"

4d) running route (while in LAN) results in: 
default         speedport.ip    0.0.0.0         UG    0      0        0 eth0
link-local      *               255.255.0.0     U     1000   0        0 wlan0
192.168.1.0     *               255.255.255.0   U     1      0        0 eth0
192.168.1.0     *               255.255.255.0   U     2      0        0 wlan0

4e) running nmap (while in LAN) results in: 
Host is up (0.0068s latency).
Not shown: 994 closed ports
PORT     STATE SERVICE
21/tcp   open  ftp
23/tcp   open  telnet
80/tcp   open  http
515/tcp  open  printer
631/tcp  open  ipp
9100/tcp open  jetdirect


If you need more info or more test results please tell me.