I just skimmed through the Jandy Heater Manual you linked to.
Since the Heater has an automatic flow control valve, the burner probably doesn’t modulate at all.
I’m thinking if the supply water is below the Set-Point Temp, the burner fires at full capacity, until the supply water reaches the Set-Point Temp. It’s either ON or OFF.
I’m sure you already knew this… but I didn’t (I run a solar heater on roof)
I’m guessing the flow control valve directs the proper flow rate through the heater to maintain a Temperature Rise in Table 6 ( 27-40 degrees F). This heated water is mixed with the Bypass water for a final temperature rise of only a few degrees for the entire flow rate.
This means you have a decent shot at measuring the Exhaust Stack Temp to confirm it’s relatively stable while the heater is firing. A Stack Temp may be as good as an On/Off signal. You could read your Natural Gas Meter before and after a 1 hour test and know how much gas the burner actually uses per hour, assuming no other major Gas Appliances run during that hour. I think this will remain constant.
Or calculate based on the heater size.
1,032 BTU per cubic foot of Natural gas (cf)
Natural Gas is billed as ccf (100 cf) so a 125,000 BTU heater would burn 1.21 ccf/hr = $9/hr.
But just like any other device… it rarely matches the actual rating. I wouldn’t be shocked it it was less than half the rating.