You are interpreting the rules correctly, but a bard of valor can cope with the restrictions.
We had a bard of valor in a campaign I DM’d for, and we stuck to the rules, which you described accurately. The bard was able to cope with these restrictions, although it required a little more “power gaming” than some players might enjoy.
Verbal-only Spells
Spells with just a Verbal component, such a Healing Word got special consideration. Healing Word is a good fit for a front-line healer. Casting Dissonant Whispers while within melee reach was another go-to tactic, letting the bard and his companions make opportunity attacks.
And the signature bardic cantrip, Vicious Mockery, gave our bard the ability to attack at range, while keeping his melee weapon and shield equipped.
Spell components are relevant
As you mention, War Caster is a great feat for a bard of valor, and our bard eventually attained that. At that point, Verbal-Somatic spells joined the list of spells that deserved extra consideration.
Juggling Sword, Board, and Focus
If the bard wanted to cast an attack spell with a material component in combat, he could drop or sheathe his sword (as a free object interaction) and to access his arcane focus.
The bard typically sheathed his sword for VS spells, or dropped it for spells with material components. If he dropped it, he would draw a spare weapon next round. (Dropping the sword became a less attractive option, though, after the bard acquired a magic sword.)
If an opponent provoked an opportunity attack while the bard had no weapon in hand, the bard still had the option to kick as an Unarmed Strike (see PH 195) and shield bash as an improvised weapon.
In some situations, such as fighting a spell caster with weak melee attacks, the bard would drop his shield and keep his sword ready for opportunity attacks. (But note that “doffing” a shield requires an action.)
I want to stress, though, that the juggling described above was not a constant thing. If a spell is cast before engaging in melee, for example, not having a weapon drawn was no issue.
A fine point about whether an arcane focus needs to be “interacted” with
To use a material component, “a spellcaster must have a hand free to access a spell’s material components — or to hold a spellcasting focus” (PH 203). But unlike “drawing or sheathing a sword,” (PH 190) accessing a material component is not explicitly mentioned as something that requires your interaction “with one object or feature of the environment for free.” (PH 190)
Whether the rules require your free object interaction to cast spells with a material component is beyond the scope of the question, but for our purposes, just find out how your DM will rule. If they rule that holding your focus does not require your free object interaction, then you probably won’t ever need to drop your weapon or shield.
Other Bard Powers
A bard often has a fair deal of Bardic Inspiration to hand out in combat, which does not require a free hand. While these aren’t spells, they are part of a bard’s action economy and their pool of limited-use resources.
Non-Combat Spells
One final point, many bard spells are non-combat spells, or have a long enough duration that they can be cast ahead of combat (e.g., Enhance Ability).
Complexity, not Disadvantage
A bard of valor may require a little more power-gaming, in terms of character build and tactics, than some character builds. But sword-and-board Rangers face similar constraints. These constraints don’t make bad character builds.
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.
Best Answer
Divergent Persona doesn't affect spellcasting
The changeling's Divergent Persona trait says (WGtE, p. 61; emphasis mine):
Note the term "ability check" at the end? Spellcasting does not generally involve ability checks. If you are casting an attack spell like Fire Bolt, it is an attack roll, not an ability check. The save DC for your spells is also unrelated to any kind of ability check.
My reading of the ability is that it would not work even with spells that can use ability checks such as dispel magic or counterspell. While both of these do allow for an ability check, if you are trying to counter or dispel a spell cast from a higher spell slot, neither allow you to add your proficiency bonus to the roll. You can add half your proficiency from the Jack of All Trades feature from the bard class, but that is not related to the instrument selected in the Divergent Persona trait.