I’ve found this:
Standard Range+ (or just "Model 3" from late 2021)
BT35 which is the Panasonic 53kWh battery with 2170 NCA cells, coded 1/1C
BT41 which is the Panasonic 55kWh battery with 2170L NCA cells, coded 1L
BTF0 which is the CATL LFP55 55kWh battery with Prismatic LFP cells, coded 6C
BTF1 which is the CATL LFP60 62kWh battery with Prismatic LFP cells, coded 6L
Long Range and Performance models
BT37 which is the Panasonic 53kWh battery with 2170 NCA cells, coded 1/1C
BT38 which is the LG Chem M48 75kWh battery with 2170 NMC cells, coded 5/5C
BT42 which is the Panasonic 82kWh battery with 2170L NCA cells, coded 3L
BT43 which is the LG Chem M50 79kWh battery with 2170 NMC cells, coded 5L
And it goes on to say
MT315 Long Range AWD
Produced: Fremont
Battery BT37 between VIN 815k and 880k (Oct 2020 and Dec 2020) for NA
Battery BT37 between VIN 2/15k and 2/98k (Jul 2021 and Nov 2021) for NA
Battery BT37 between VIN 815k and 917k (Oct 2020 and Mar 2021) for Europe
Battery BT37 between VIN 815k and 995k (Oct 2020 and Apr 2021) for APAC
The battery was the BT37 and built between Oct 2020 and Nov 2021 in Fremont. VIN numbers range from 815k, go through the VIN roll over and on to 98k. The cars were distributed to all markets, the US mainly until Dec 2020 / VIN 880k, with a handful between Jul and Nov 21. Europe and APAC had deliveries until March /April
The EPA range was 353 Miles or a WLTP 360 Miles/580km. It could do 0-60 mph in 4.2s, 0-100km/h 4.4s and had a top speed of 145mph or 233km/h
MT316 Long Range AWD
Produced: Fremont
Battery BT38 between VIN 817k and 944k (Oct 2020 and Apr 2021) for Europe
Produced: China
Battery BT38 between VIN 164k and 418k (Jan 2021 and Nov 2021) for Europe
Battery BT38 between VIN 164k and 359k (Jan 2021 and Aug 2021) for APAC
The battery was the BT38 and built in both Fremont and China. Fremont production was from Oct 2020, VIN 817k through to Apr 2021, VIN 944k and went to Europe. China production was between Jan 2021 and Nov 2021, VIN numbers range from 164k through to 418k and went to Europe. APAC also had China produced cars although these stopped in Aug 2021. No cars went to NA.
The WLTP range was 360 Miles/580km. It could do 0-60 mph in 4.2s, 0-100km/h 4.4s and had a top speed of 145mph or 233km/h
MT321 Long Range AWD
Produced: Fremont
Battery BT42 between VIN 900k and 2/149k (Feb 2021 and Jan 2022) for NA
Battery BT42 between VIN 900k and 973k (Feb 2021 and May 2021) for Europe in LHD
Battery BT42 between VIN 900k and 973k (Feb 2021 and May 2021) for APAC in small numbers.
The battery was the BT42 and built in Fremont. Production was from Feb 2021, VIN 900k through the VIN roll over and on to Jan 2022, VIN 149k, although few cars seem to have been built after Oct 2021 and VIN 2/94k. Most cars went to NA, with some LHD cars going to Europe up until May 2021 and VIN 973k. A handful of cars also found there way to APAC.
The EPA range was 353 Miles/568km, the WLTP rating was 614km. It could do 0-60 mph in 4.2s, 0-100km/h 4.4s and had a top speed of 145mph or 233km/h
MT323 Long Range AWD
Produced: China
Battery BT43 between VIN 383k and 345k (Oct 2021 and Nov 2021) for Europe
Battery BT43 between VIN 383k and 345k (Oct 2021 and Nov 2021) for APAC
The battery was the BT43 and built in China and was fairly short lived. China production was from Oct 2021, VIN 383k through to Nov 2021, VIN 345k. Cars went to Europe and APAC in both LHD and RHD form.
The WLTP rating was 614km. It could do 0-60 mph in 4.2s, 0-100km/h 4.4s and had a top speed of 145mph or 233km/h
MT324 Long Range AWD
Produced: Fremont
Battery BT42 between VIN 2/100k and 2/392k (Nov 2021 and Nov 2022) for NA
The battery was the BT42 and built in Fremont. Production was from Nov 2021, VIN 100k through to Nov 2022, VIN 392k, although most production stopped in Sept 2022 at VIN 350k. Cars went to NA, with a few cars going to APAC.
The WLTP rating was 614km. It could do 0-60 mph in 4.2s, 0-100km/h 4.4s and had a top speed of 145mph or 233km/h
MT328 Long Range AWD
Produced: China
Battery BT43 between VIN 430k and continues (Nov 2021 onwards) for Europe
Battery BT43 between VIN 430k and continues (Nov 2021 onwards) for APAC
The battery was the BT43 and built in China. Production was from Nov 2021, VIN 430k and continues. Cars went to Europe and APAC.
The WLTP rating was 374 miles. It could do 0-60 mph in 4.2s, 0-100km/h 4.4s and had a top speed of 145mph or 233km/h
There looks to be lots of versions depending where you are but I think you can find your MT3xx number by looking at the html on myTesla once you’ve selected your car.