Reverse-engineering a secure sensor protocol is a lot of work.
I would look at the transmitter circuit design. Sometimes they bungle it as a bad design, or intentionally weaken TX power for FCC approvals. U2 is a
F4455B ASK transmitter in SOT23-6, your circuit is missing C4 maybe check the circuit a bit.
First I'd try put a decent antenna on it. Or look for what's blocking /interfering with the RF signal.
You could tap into the keying input and then just add your own high power TX.
IC datasheet:
12.1 Use the power amplitude control of external resistance
The R7 is used to adjust the RF amplitude output level, which may be required to meet consistency adjustment. For example, the following table lists a 50 ohm test
The conductive RF output level of the test board and the typical value of the corresponding R7 resistance value, as shown in Figure 2. F4455B demonstration board using ring antenna R7 can adjust the radiation field allowed by FCC or ETSI compliance as needed. Please contact Sinota to get the recommended R0 value, so as to Comply with FCC and ETSI standards."
Oh well, there is no R7 or TX power table in the datasheet. Must be on the eval board.