I would do the bouncing time measurement as follows:

Start the measurement by activating "DRV1", for instance. Make, that the DUT is turned-on, when "DRV1" is activated.
Have the switch contact of DUT connected with one side to ground and other side to 5V, via pull-up of let me say 499R, according to a switching current of 10mA. Don't use the 10mA current source of card, because during fast changes of load, like it is expected during the bouncing, means with a highly dynamic load, current source can behave strange.
Connect this contact side (which is pulled-up) to the "TRIGGER" line, which goes to the input of U15A, a 74LS125.
Immediately after the DUT was turned-on, make a series of successive readings of "D1" by activating "SW_CLOSE", which gives you the potential of output of U15A. As reed relays are rather fast you should read "D1" every 10µsec.

What will you observe now?

The first 10µsecs, nothing. Because the contact tongues need a certain time to "travel", due to inertia of mass. So, "D1" is "0" during this turn-on delay.

After some 100µsecs the contacts will make the first switch, so you will read D1 = 1 for the first time. This moment should be the starting point of your bouncing time measurement.

Due to bouncing the contacts will open and close again repeatedly up to the end of bouncing. So, you will see 0 to 1 and 1 to 0 togglings at "D1" during the bouncing.

The final 0 to 1 toggling at "D1" marks the end of bouncing.

The turn-on time is standardly defined as the sum of turn-on delay and bouncing time. So, this measurement allows you to determine three parameters: Turn-on delay, turn-on bouncing time and turn-on time. A typical turn-on time of a reed relay is about 0.5msec.

Now, once having turned-on the DUT, you can measure the turn-off parameters by deactivating "DRV1" again. The according measurement will give you the turn-off delay, the turn-off bouncing time and the turn-off time. A typical turn-off time of a reed relay is about 0.2msec.

Kai