It seems like you are in deep water with a client insisting on influencing technical solutions that should be up to the sound person to handle. I would probably simply hire a professional for the job.
The standard solution for this kind of setup would be to use a sound mixer with at least two inputs. Use one channel for the receiver part of the wireless lapel mic. Use next channel for the wired podium mic, connected with a XLR cable (which can be upwards to 300 feet if needed).
Using a mixer allows you to have several different inputs and set their volume levels separately.
If the client really demands it you could of course have a wireless podium mic, but it can often create more problems than it solves.
Cost wise, a mixer could start at USD 100 although I much preferr a bit more ruggedized stuff. A podium mic, say a Shure SM58 with a mic stand or similar holder + XLR cable could start at USD 200.
The wireless lapel can be more or less expensive. The low priced ones today often sends on WiFi band and can as noted be disturbed by high WiFi usage. Examples could be the Rode Wireless Go (you will need an adapter at the mixer end). Another solution is a radio frequency mic from the likes of Sennheiser or Shure, these are less sensitive to WiFi disturbances but come at a higher price. The professional level systems can be used with directional antennas (shark fins) greatly reducing interference from other sources even with on a longer distance. A specific problem is that allowed freqency usage varies between countries.
Whish you good speed.