Based on schematic DSD Tech points to for their HC-06 product (minus the RS232 converter):
DSD TECH BT-06, the entire module is power by a 3.3V regulator. However, there is a 10k Ohm pull-up resistor to 5V on the module TX output. This will limit T4.0 damaging current. So, maybe only the T4.0 GPIO pin is damaged. HC-06 TX output should not be damaged although this is what you observed with Low around 3.5mV.
5V applied to T4.0 Vin is correct.
I suspect you have a soldering short circuit on HC-06 Tx signal somewhere. Solder short is either at the T4.0 or at the other end (HC-06). Let's see if we can isolate the problem.
With everything wired up as you want it, disconnect only the HC-06 TX output wire at the HC-06. Now, power everything up and measure the voltage at HC-06 TX pin and measure the HC-06 5V input pin. Be sure to measure
on the pin, not on the wires.
If you observe 5V at both of these pins under this test condition, the issue is either with wiring, soldering, or problem with T4.0.
Now, leaving everything setup as above, measure Teensy RX pin.
Let us know the results.
Can you upload a photo of T4.0 soldering and the HC-06 soldering?
BTW, you should not connect T4.0 USB to PC USB when using the ATX power supply. The PC USB supplies 5V to T4.0 on USB cable and this will fight with ATX 5V supply. If you need USB and ATX at the same time, there are a few options. See T4.0 back-side of card
<here> to locate 5V pads. Then, select an option from
<here>. The weblink shows a different Teensy, but the result is the same for T4.0. And, think of the ATX supply as if it was the battery being shown.
<edit>
Are there any components on the backside of HC-06? There aren't any photo's of backside available. The reason I ask is, it's not clear to me if the DSD Tech schematic at the weblink they provide, is representative of the module you have. For example, the schematic shows an RS232 chip. But, there's only TTL TX/RX signal connections according to DSD Tech product description (at amazon and DSD Tech websites). So, I've assumed the 10k Ohm pull-up resistor to 5V is installed on HC-06 TX output pin. This could be a wrong assumption.
With the module powered OFF and TX pin wiring disconnected, measure the resistance from TX pin to 5V pin on HC-06 module. Maybe you'll observe 10k Ohms, maybe you won't.