You just need the hand crossbow, and nothing else; the hand crossbow is a one handed weapon, and it enables its own Crossbow Expert bonus action attack. In addition, as Doval points out, PHB errata has removed the word 'loaded' from the Crossbow Expert text quoted in the question.
(Note: I do not take Colossus Slayer into account in the following analysis, as it doesn't particularly affect which weapon setup is the best.)
Turn 1, 2, and 3: Attack action for 2× attack w/ hand crossbow, Crossbow Expert bonus action for 1× attack w/ hand crossbow:
3×(d6+4) = Avg. 22.5 damage/turn, before taking hit chance into account.
This results in a total of 67.5 damage over 3 rounds. Assuming a 60% hit chance, that comes out to a realistic expectation of ~40.5 damage in the first 3 rounds.
If you were willing to expend spell slots, casting Hunter's Mark as your bonus action (instead of attacking) on turn one would bump that up to 88 damage over 3 turns, or 52.8 damage when adjusted for accuracy.
The only way I see to potentially do more damage than that would be something that somewhat violates the spirit of your conditions, if not the word. It also falls behind compared to purely using a hand crossbow as above if the Archery fighting style is chosen rather than Defense.
Carry a supply of rapiers on you in addition to the hand crossbow. For the purpose of this analysis, at least 3.
Assuming you start with the hand crossbow drawn before combat:
Turn 1, 2, and 3: Attack action for 2× attack w/ rapier, Crossbow Expert bonus action for 1× attack with the hand crossbow. Each turn, use your free object interaction to draw a new rapier from one of your excessive number of sheaths, then after finishing the Attack action, drop it for free. You can then reload/fire the hand crossbow as a bonus action since you have a hand free.
(d6+4)+2×(d8+4) = Avg. 24.5 damage/turn.
This results in a total of 73.5 damage over three rounds. Assuming a 60% hit chance, that comes out to a realistic expectation of ~44.1 damage in the first 3 rounds.
If you start with no weapons drawn, the first turn has to be done with hand crossbow only, reducing the damage for that turn back to the same 22.5 as above for a total of 71.5 damage over three rounds (or 42.9 with 60% accuracy).