When you mention router at the extended location, is that because you want to present both wired ports and Wi-Fi there?
A gigabit switch would provide wired ports without adding Wi-Fi and would be much cheaper (and nothing to configure) if you only want to plug things in.
If you do go for a 2nd router it does not need to be a DSL type as for extending the LAN you are not using its WAN port (only needed on the Fritzbox).
prlzx on Zen: FTTC (VDSL) at ~40Mbps / 10Mbps
with IP4/6 (no v6? - not true Internet)