How to find radio frequency from a "missing" key for a 2006 Mazda 5?

Is there any way to pick up a radio frequency from a key that might be lost somewhere inside a 2006 Mazda 5?

My wife lost one of our two car keys for our 2006 Mazda 5 so we were down to 1 key.
I took the car to the dealer (Carling Motors in Ottawa Canada) to get a new key made.
After $355, I walked out with a new key.
When I got to the car, I tried my original key and it would not work.
I went back and spent next 2 hours with them trying to re-program the key.
They even got me a new key so I have a 2 new keys.
However, they could never get a 2nd key programmed so only 1 works.
As the dealership was closing, they came out with their "latest" theory.
They believe that the lost original key is somewhere in the car.
(The belief is that if another transmitters is being operated in the vicinity, then they can't program the key.)
This weekend, I went through every nook and cranny and I can't find this missing key that they believe is lost in the car.

My confidence in the integrity of my dealer is at its lowest point.
However, let's give them the benefit of the doubt.
Is there any way to pick up the radio frequency that this "lost" key is emitting?
THANKS

think the dealer is mistaken.

The precaution in the workshop manual goes:
========================
Caution
If any of the following items are touching or near the key head, signal communication between the
key and vehicle is negatively affected, resulting in the engine not starting or the key registration
failure. Do not perform the procedure if any of the following items are touching or near the key
head.
Any metallic object
Spare keys or keys for other vehicles equipped with an immobilizer system
Any electronic device, or any credit or
other cards with magnetic strips

-
So unless your lost key is hanging by the ignition switch, or stuck inside the steering column. the engine cannot sense the presence of a key.

Or if they say the key is transmitting... I thought we have to push a button for it to transmit.
===============================

I assume they read the workshop manual. In the MAZDA5 2006 Work shop manual.. Pages 09-14-16 to 09-14-24. There are procedures even for
clearing out all key ids and starting fresh.

======

============ quote from the manual =================

No.2 Additional Key Registration Procedure (Using the WDS or equivalent)
Conditions
Customer has only one valid key. Or customer has no valid keys. (Can also be performed even if there are two
or more valid keys)
Note
A maximum of eight keys can be registered for any one vehicle. If key registration is not successful and
DTC 15 appears even though the procedure was performed properly, use the PID/data monitor function of
the WDS or equivalent and verify the number of keys that have been registered.
If eight keys have already been registered, and it is necessary to register other keys, the previously
registered key ID numbers must first be cleared. To clear the key ID numbers, refer to 091421 No.3
Key Replacement Procedure (Clearing Previously Registered Key ID Numbers, Key Re-registration).
Procedure
1. Have one key (key 1) ready for registration.
2. Using key 1, turn the ignition switch to the ON position.
Note
Although the security light flashes and DTC 15 is displayed, this does not indicate an improper procedure.
Continue to perform the procedure as indicated.
3. Connect the WDS or equivalent to the DLC-2.
4. Select Body/Security/PATS function from the WDS or equivalent screen menu.
5. Perform security access as indicated on the WDS or equivalent screen. (See 091424 No.6 Security Access
Procedure.)
6. Select Program Additional Ignition Key from the WDS or equivalent screen menu.
=================

They should solve the problem. Because in effect, they have 'disabled' your
original key. and programmed you a new one. What if you find your 'lost key'? That one will also not work anymore I assume.