Technical Question: Effect of using TOSLINK to connect PC to BT Transmitter to BT AMP/DAC

I’m setting up an amateur wireless “hifi” headphone system. I am faced with quite a few questions arising from my set up:

(1) If a Bluetooth audio transmitter is only connected to the Laptop by means of an Aux cable, does that mean that the audio passing through the Laptop down to the BT Transmitter already underwent a digital to analog conversion in the Laptop’s own DAC? And consequently, will the audio to be transmitted by the BT Transmitter to a BT Amp/DAC be an adulterated digital audio (considering that Bluetooth transfers audio digitally) by reason that it had already been previously converted in the Laptop’s DAC?; and

(2) If a Bluetooth audio transmitter is only connected to an external soundcard of a Laptop through a TOSLINK cable, does that mean that the audio passing through the Laptop down the BT Transmitter would still be an unadulterated Digital Audio format (since TOSLINK transfers data digitally)? And consequently, will the audio to be transmitted by the BT Transmitter to the BT Amp/DAC still be an unadulterated digital audio? Thus, will this result for the BT Amp/DAC to convert an unadulterated digital audio format into analog for the headphone?

I know it’s crazy, but to add some details, I am planning to have this set-up: Laptop → BT Transmitter → Fiio BTR3K → Philips SHP9500. Since my laptop does not support aptx, I am compelled to use a BT Transmitter. Now, my budget’s tight for a BT Transmitter (I’m still saving up for that Fiio BTA30), and the best value that I can find are some UGreen BT Transmitters which supports apTx HD and LL. The thing is, it comes in 2 forms, either an AUX cable or a TOSLINK cable. Then, that goes to my question. Thanks!