Ok, when you mentioned WiFi I thought that you were using an external WiFi module. The RFM69HW is a radio transceiver in the UHF amateur band and does not use WiFi frequencies which are much higher.
You won't be able to connect a multimeter to the RFM module without desoldering the modules from the board, so better not to.
When module starting to transmit could cause a surge in power that the converter takes some time to respond to. Introducing a delay to the transmit would indeed be a good solution.
Do you see the fluctuation of the sensors with the qbcan example code?
Also, could you explain a bit more the fluctuations? like we are talking 0.1ºC fluctuations every second or more like 5ºC?very small fluctuations (0.1-0.5ºC) are expected as the power supply is not perfectly stable and BMP180 sensor characteristics.
Bear in mind as well that the board does heat up a bit when being used just due to the components present not being 100% efficient.