Yes, the way you did it, you just lose this expertise.
Okay, let's start with some background.
Do expertise bonus stack?
It seems they don't. But what doesn't stack is their bonus.
RAW, it seems you can put your expertise twice in the same skill. It just won't stack the bonus (therefore there is no reason to do it).
Let's check why I say we can do it:
At 1st level, choose two of your skill proficiencies, or
one of your skill proficiencies and your proficiency with
thieves’ tools. Your proficiency bonus is doubled for any
ability check you make that uses either of the chosen
At 6th level, you can choose two more of your
proficiencies (in skills or with thieves’ tools) to gain
At 3rd level, choose two of your skill proficiencies. Your
proficiency bonus is doubled for any ability check you
make that uses either of the chosen proficiencies.
At 10th level, you can choose another two skill
proficiencies to gain this benefit.
Note that in the 1st Rogue level and 3rd Bard level, there is no "another" or "more" or "other" skills. Obviously, because so far they aren't considering Multiclassing (as it is a specific rule). You would expect to see some clarification about this in the Multiclassing section, but there is none.
RAW, you can get expertise twice in the same Skill, but it does not stack.
So yes, you just "lose" this expertise.
Quick Remark about the Multiclassing you did
Note that if you go the other way around multiclassing, starting with rogue, you will have 7 skill proficiencies (4 from rogue + 1 from bard + 2 from BG) and a thieves' tools, so you can actually use all your expertises.
A great comment added by Giuseppe added some more solutions:
- You can take a race that grants a skill proficiency, such as elves.
- As you mentioned in your question, you can get a feat that grants a skill proficiency, such as Skilled.
- Instead of taking 10 levels in Rogue, since you get the max expertise at 6th level, you could instead multiclass into Ranger, which would give you an extra Skill Proficiency. Side Note: By PHB Ranger is the only class that you could still get more Skill Profs.
RAW, you lose 1 point of expertise. RAI, you probably lose 1 point of expertise. But you can still build your skill monkey by starting your multiclassing with Rogue instead of Bard. Or you can just talk to your DM.
Side note: Fun concept you got there to get to this question.