You could more cheaply and possibly more effectively connect a WAP to your existing router.
Re adding a second router, you don't normally do that the way you suggest, though it would work. And yes, you would lose the ISP login details which would need to be in the new one.
The normal thing people do is disable DHCP in the new router and adjust its own default LAN address to one on the original router's LAN. Making sure you give it one outside the original one's allocatable range, to avoid the possibility of two on the LAN with the same address.
As a bonus, both these solutions give you two access points, whereas yours would leave you with just one. A portable device such as a phone, tablet or laptop can move almost seamlessly between the two if walking around with it.
The indispensable man or woman passes from the scene, and what happens next is more or less the same thing as was happening before.
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