Warning: I do not have a ATtiny85 nor the lcd so this is purely based on what I found after some reading up on this mcu.

What I would do in your situation: As the trinket is using a ATtiny85 it does not have hardware serial. So a software variant is needed.
You opted for software serial (which is the most obvous in Arduino world) but which seems to be the hardest solution. According to this source enter link description here is is possible (And I quote)

you might have to tune the internal oscillator to get a stable connection.

Assuming your lcd does not need to send data back; I would try the TinyDebugSerial.

Warning: I do not have a ATtiny85 nor the lcd so this is purely based on what I found after some reading up on this mcu.

What I would do in your situation: As the trinket is using a ATtiny85 it does not have hardware serial. So a software variant is needed.
You opted for software serial (which is the most obvous in Arduino world) but which seems to be the hardest solution. According to this source Serial communication with the Tiny's is is possible (And I quote)

you might have to tune the internal oscillator to get a stable connection.

Assuming your lcd does not need to send data back; I would try the TinyDebugSerial.