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In our church orchestra we play three selections and head out to the pews for the service. About 45 minutes later we return to our seats for a final song and/or any specials.

With my 3/4 Jupiter I would stand the horn bell end down on a carpeted floor. Now, I will be playing a much larger Yamaha 201. The case is in the green room, too far to walk for tis interval. I can lay the tuba on the edge of the stage or place it bell down by my seat as I have been with the smaller horn. At east two drummers, three trumpets, and a French horn player have to walk by my seat to get to theirs. I let them pass and follow them into the pit.

After we are done, I will carry it backstage to the green room and put it in its case.

If it was YOUR tuba, where would you place it between songs?

This of course depends on the setting of your platform. Nobody walks by where I set my horn and I place my horn on it's bell as well. The only person who might walk near my horn sits to my left and he usually exits to the left. I set my horn down to my right, where as I said, nobody walks by

In the case, as many members of my church were once viola players.

In my mancave/tuba room...because I don't play in your church's orchestra.

Buy a K&M stand

Seems most simple to simply change the seating so that you are farthest back.

REAL ANSWER:

Leave the pit BEFORE all of those people who pass by your chair...(or hold it vertically in your lap, if they MUST exit before you).
Re-enter the pit AFTER all of those other people...((or hold it vertically in your lap, if you MUST enter before them).

I use a K&M stand in which the tuba is upright (bell pointed up). I prefer the bell up so others can't stand on the bell. This way the bell creates also a kind of barrier for others walking "around" the tuba. Not that others walk around my tuba, because i will scare 'm away with my angry face .
When my tuba stays behind and we (band/orchestra) leave, I will be there till I'm confident that there is no more traffic in the direction of my tuba and be back before the returning traffic starts.

Simple answer really, never leave your Tuba unattended for periods of time when other folk might pass it. Leaving it unguarded is next to asking for someone to damage it. The OP might need to ask for a change in seating plan and other stuff to ensure his instrument’s safety, nobody intends to have an accident but they still happen and the OP will end up picking up the bill for the repair work. Music is important to worship and sometimes you have to accept risks so that you can support a service, on the other hand if a satisfactory arrangement can’t be reached then I’d feel guided to letting the music group get by without a Tuba until a solution was presented.