Not sure if this question deserves a new thread, but as it is connected to this one, I'll ask here for now.

Just as MaxwellSheffield did, I also would like to have an idea about the correctness of the pricing for an option on a STIR futures. To make the example concrete, let's say the call@96.75 on the September 2026 on the SOFR contract.

Bloomberg at the moment gives me the following quote (SFRU6 Comdty OMON):
Ticker: SFRU6C 96.75
Days:340
Rate: it uses 4.06%
Futures level (Sep26): 96.865
Last: .3525y (the "y" means that the quote is for yesterday's close)
IVM: 22.63 (I have the setting in OMON to use Black76).

Now, my problem here is that if I use Bloomberg itself, and I type in "ThPx" (theoretical price) the price of the last quote 0.3525, then the implied volatility it returns is: 78.48, completely off from the IVM quote.

My next step was to price the option myself with a Black76 calculator.
If I plug the following:
Option Type: "P" (if the option on the 100-rate is a call, the option on the rate is a put)
Forward: (100-96.865)/100 = 3.135%
Strike: (100-96.75)/100 = 3.25%
Time to expiry: (340/365) = 0.931507
Volatility: 78.48%
Then I get a fwd premium of: 1.000643. The discount factor should be in the order of: exp(-4.06% * 340/365) = 0.962887
so the PV should be 1.000643*0.962887 = 0.963506, very different from either of the 0.3525 I started with on the BBG terminal.

Can you see what I am doing wrong? IF you do, would you tell me please?

Thank you in advance
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Update 1

I discovered the Options Calculator on the CME Group website and the pricer there seem to validate the calculation. Now my impression is that BBG somehow mixes log normal (IVM) and normal (IVol), so IVM is always log normal, while IVol is log normal when it shows the feed from BBG but it becomes a normal vol quote when one overrides the price. This feels very weird, but I'll check with BBG support.

I guess that the residual question is whether the quotes from BBG can be trusted (not the vols but the premiums)... and for which strikes...  Does anyone has any suggestion? Are all the option prices equally tradable (I would suspect they aren't). Which minimum open interest would make the options on SFR real tradable market quotes? Thank you!

Update 2
Making progress understanding the data... so once the pricing question was cleared with the CME option tool, I could make sense of the data on Bloomberg. 
I wouldn't use the implied vol from BBG as they mix current quotes with quotes from previous days, so I found it noisy, but provided I worked with the premia themselves, I can see they match the CME website. More confusion came from using BQL() instead of BDP() where BQL() was giving me the volumes from previous days if no volume was available for today (which incidentally makes me very wary of using BQL going forward), but the price data is fine.
So far my conclusion is to ignore all options trading at the minimum price (1/4th of bp) because they are almost certainly worth less than their minimum value (and adding them in any skew calibration increases the implied volatility at the wings).
Then, depending on the task filter the option based either on open interest or volume as taking the whole set (with BBG) mixes prices from today and previous days, and that also adds a lot of noise.

And... now I have another question about pricing... so I'll post it as a separate post... 

Hopefully my last question about options on STIRs...

When I did some work last time (before this week) about 20 years ago, I think (I may remember incorrectly) that I was using Black76 but without discounting the premium and the reason was that the premium isn't paid today but as it is a margined instrument, the option was bought through margining requirements, so there was no initial payment today and it was therefore correct to consider the premium as a premium paid at maturity (hence interest rate in Black76 set to 0).

1. Was I wrong and I should have discounted the forward premium to today instead?
2. If I wasn't wrong, why does the CME Option pricer discount the forward premium? Is there a fundamental difference in this regards between the old USD options on LIBOR 3M futures and the current on SOFR 3M futures?
3. Again in the olden days, the interest rate was positive and there is no "dividend" payment for a STIR therefore it was never optimal to exercise a call option on a STIR future earlier, and based on this we justified using Bachellier/Black76 instead of an american model (I remember implementing Barone-Adesi back then for those). Does this still hold with SOFR? A trader told me that there is now some advantage to exercise early and I am wondering why this is the case...