Since I have nothing better to do I tried to look into any FCC filings for the Joy-Cons. If HAC-015 (
https://fccid.io/BKEHAC015) is the left Joy-Con, which seems likely, then we already have some information about the maximum output power.
This suggests the output power is around 3-4 dBm, which is in-line with what you would expect from a Bluetooth Class2 device (maximum output power of 4dBm). While Bluetooth technically allows up to 20dBm output power for Class1 devices, most chips today will allow up to 10dBm output power. This is informally known as Class1.5 and is common since FCC does not require SAR testing for any devices with an output power below 10dBm.
The good news is then that Nintendo will most likely be able to increase the output power by around 6dB, which should in theory give around twice the current range. This will also not negatively affect the Battery life significantly, as the chip will dynamically change the output power as required. So the additional power will only be used when it otherwise would lose the connection.
The bad news is that the Joy-Cons will have to be re-certified, which means that Nintendo will likely not be able to have this done as a day-one firmware update unless they already caught this issue ahead of time and are in the process of fixing it already. (They will also likely have to measure SAR as well even if they are below the 10dBm limit, since Canada recently lowered the limit where you do not have to measure SAR to 6dBm).
The thing that surprises me however is that 4dBm should absolutely be enough power for the type of environments shown in the video above. Deliberately covering the entire device with both hands will probably always be a problem. The antenna is probably quite narrowband so doing that will likely detune it as well causing even more losses than what you get anyway from the absorption in the hands. But simply placing the device behind the back and breaking line-of-sight? When you are indoors and only a few meters away from the main console, this should absolutely not be a problem. So unless there is something really strange going on with their antennas (like if they didn't account for the detuning that might happen when you hold the device normally) then it might actually be possible that the real Joy-Cons are using lower power than they are certified for, either due to a firmware bug or because someone though it would increase battery life (it really shouldn't). In that case, it would be fairly simple to issue a new firmware to increase the power.
