You can
Psychic Blades work like this (TCoE, p. 64):
Whenever you take the Attack action, you can manifest a psychic blade
from your free hand and make the attack with that blade. [...] The blade vanishes immediately after it hits or misses its target, and it leaves no mark on its target if it deals damage.
That is, your hand has to be free to use the feature, and you manifest the blade as you make the attack. ("manifest ... and make the attack"). You cannot use the feature if you hold a weapon in your hand.
You could hold a weapon in your other hand while you manifest the blade, there is no restriction against that. If you then pass that weapon to your original hand, you later could use the bonus action to manifest and use another blade in that other hand, because it now is free. The only requirement is that you attacked with the first blade, and that the bonus action comes after that.
The bonus action can be used at any time on the same turn after the attack with the first blade, it is not immediately after the first attack, or when you make the first attack. It also does not require that your other hand was free when you made the first attack, only that it is free when you use it.
That means you could pass the weapon from your other hand were you first held it to your hand with which you used the feature first with a free object interaction, so action economy wise, this should be no problem.
Comparison to two-weapon fighting
This is different from two weapon-fighting (p. 195) that says
When you take the Attack action and attack with a light
melee weapon that you’re holding in one hand, you can
use a bonus action to attack with a different light melee
weapon that you’re holding in the other hand.
Here, the condition is that you have to hold both weapons at the same time. This has been confirmed by Sage Advice
Can I make an attack with one weapon, then draw a
second weapon with my other hand and qualify to use
two-weapon fighting? To use the two-weapon fighting
bonus action (PH, 195), you must have both weapons in
hand when you make the first attack.
The language is slightly different. Still, to be safe I would check with my DM. It may be that they rule this should work like two-weapon fighting in that both your hands need to be free to use two psychic blades.