Reading the feature more carefully and "Casting Spells at a Higher Level," it's clear that this is not possible (despite my hopes):
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. For instance, if Umara casts magic missile using one of her 2nd-level slots, that magic missile is 2nd level. Effectively, the spell expands to fill the slot it is put into.
That is, casting at a higher level is predicated on using higher-level slots (which is not the case here). Put differently, there are 3 elements: the level of the spell (which the Psionic Sorcerer references, in singular), there is the level of the slot, and by using the higher slot, there's the level the spell assumes (i.e., that of the higher-level slot).
However, without the use of a higher-level slot, the spell can only assume its innate level. A L1 spell can only become a L3 spell by casting it with a L3 slot (but at that point, it's already cast—you can't use 3 SP to cast the L3 version).
There is no rule/allowance for up-casting with resources other than slots unless otherwise indicated (confirmed in Sage Advice):
Sage Advice Compendium version 2.3, page 13:
What level is a spell if you cast it without a spell slot?
Such a spell is cast at its lowest possible level, which is the level that appears near the top of its description. Unless you have a special ability that says otherwise, the only way to increase the level of a spell is to expend a higher-level spell slot when you cast it. (p 13)
Contrast with the Monk feature, Disciple of the Elements, discussed in the Sage Advice answer:
"The Monk’s Disciple of the Elements feature lets the monk spend ki points, rather than a spell slot, to increase the level of a spell."
Disciple of the Elements:
"Once you reach 5th level in this class, you can spend additional ki points to increase the level of an elemental discipline spell that you cast, provided that the spell has an enhanced effect at a higher level, as Burning Hands does. The spell's level increases by 1 for each additional ki point you spend."
The Psionic Sorcery feature does not mention anything like that.
More details here.