As a fretless bassist, I'm interested in your quest and do not envy your position. Since my chosen instrument is neither fretted nor upright, it also doesn't fit the mold for music teachers.
Maybe, being in the teacher's chair magnifies the difference between 4 & 5 string basses and he's not comfortable in renovating his curriculum (and manuscripts) to suit a minority instrument - especially sight reading with revised scale and arpeggio fingerings. If the five string bass is your chosen instrument, then you need a teacher who wants to teach that instrument.
I took six music lessons when younger with my fretless bass. The teacher was a guitarist, well known in the city. After two lessons he decided, selflessly, to turn me over to a double bass player because he thought I would gain more from him than from a guitarist. Well, I wish he would have stuck with me, because the double bass teacher spent most of the time yacking or bragging that the only bass was an upright bass.

And his fingering was different from the guitar fingering used by bass guitarists.

This is when I realized that I am not a bassist - I am a "Bass Guitarist" - and this inspired me to learn to play guitar also.

I never renewed, to my loss, as I much preferred the older guitarist - a real, experienced musician/educator who coaxed me to sing along as I played and to write out the scales myself. His chording along when I played bass was very inspirational.
You may consider not to give up on a good teacher and replace him with some amateur who may waste your time and money.

There are hundreds of years of skill and knowledge amassed on the four string instrument.
Would you consider buying a really cheap 4 banger to take your lessons with a good teacher and a proven curriculum?

Or keep searching for a dedicated teacher who has the music and fingering for the fiver? And what are your goals? At the moment, all music is written for four string double bass and bass guitar.

This leaves that extra string to hang there unplayed and resonating, except when your playing for fun. And where does it fit into conventional music gigs? I don't envy your position, but it's a nice position to be in.
Thanks for starting an interesting thread. Cheers!
