Yes, it can

You can make any non-magical trinket that can fit in your hand, and it disappears after the end of your next turn. One of the trinkets you can make, as listed under PHB 160, is "a whistle made from gold-colored wood" as well as "a tiny silver bell without a clapper"; these are both things that you could make such that they can fit in your hand.

Furthermore, they fit the criteria of "musical instruments," because you can make music with these. While the PHB does list some of the things it considers as instruments, it is not meant to be an exhaustive list:

Tools, PHB 154

Musical Instrument. Several of the most common types of musical instruments are shown on the table as examples.

This means prestidigitation can create a musical instrument that can fit in your hand, and you can use any musical instrument as a spellcasting focus.

Of course, it is inefficient

You must use your action to cast prestidigitation, and then you cannot use your action again to cast another spell. On your next turn, you could use the instrument to cast a spell, but then it disappears at the end of that turn, and you have to recreate it on your third turn. This allows you one spell every other round that isn't prestidigitation (unless it's a bonus action or reaction spell).

Reference the tool proficiency optional rules from Xanathar's Guide to Everything

You can't apply your proficiency bonus more than once to a single roll, as you've established; it seems like you should probably make this roll using your expertise in performance, since performance is a relevant skill and with expertise that grants you the best modifier. However, there are options to represent your particular combination of proficiencies beyond simply adding more bonuses to the roll.

Xanathar's Guide to Everything, in the Dungeon Master's Tools chapter, includes a section titled Tools and Skills Together, which addresses exactly this situation where a character might have overlapping tool and skill proficiencies. It makes a couple of suggestions about how a DM could adjudicate the situation:

Advantage. If the use of a tool and the use of a skill both apply to a check, and a character is proficient with the tool and the skill, consider allowing the character to make the check with advantage.

[...]

Added Benefit. In addition, consider giving characters who have both a relevant skill and a relevant tool proficiency an added benefit on a successful check. This benefit might be in the form of more detailed information or could simulate the effect of a different sort of successful check.

The chapter then goes on to describe the possible benefits of various tool proficiencies, and gaining advantage on a performance check incorporating that instrument is given as a benefit of proficiency with a musical instrument:

Skills. Every tool potentially provides advantage on a check when used in conjunction with certain skills, provided a character is proficient with the tool and the skill.

[...]

Performance. Your ability to put on a good show is improved when you incorporate an instrument into your act.

An extra benefit to a successful check might perhaps be allowing the bard an automatic success (or at least granting advantage) on a subsequent social skill check to influence the audience after the performance.

The intent of these optional rules from Xanathar's is to make tool proficiencies more valuable, in order to encourage players to take and make use of them as character options, and reward characters who have invested in such specialisation.