Can I use the Soulknife rogue's Psi-Bolstered Knack option (from the Psionic Talent feature) with the artificer's Flash of Genius feature to add both my Intelligence modifier and my Psionic Talent die to the roll?
You can use them both but you have to fail first
Flash of Genius reads in part:
When you or another creature you can see within 30 feet of you makes an ability check or a saving throw, you can use your reaction to add your Intelligence modifier to the roll.
This is pretty straightforward. You burn your reaction to add your intelligence modifier to the check.
The Soul Knife flavor of rogue's Psi-Bolstered Knack reads in part:
if you fail an ability check using a skill or tool with which you have proficiency, you can roll your Psionic Talent die and add the number rolled to the check, potentially turning failure into success.
The critical wording here is that you must first fail the check in order to use the feature. However, if you fail the check even with the additional modifier from Flash of Genius, then yes, you may retry with the die provided by your Psionic Talent Die.
But can you use the Psi-Bolstered Knack first and then, if you still fail the check, use Flash of Genius?
The answer depends on how you interpret the rules. There are abilities that allow the player to see the roll result before deciding activate them and add modifiers such as Bardic Inspiration:
The creature can wait until after it rolls the d20 before deciding to use the Bardic Inspiration die, but must decide before the DM says whether the roll succeeds or fails.
Other examples where the player is explicitly allowed to see the roll result before deciding to use the ability include Wild Magic Sorcerer's Bend Luck and War Domain Cleric's Guided Strike.
Because Flash of Genius does not include language that allows the player to "peek" at the result before deciding to use it, I would tend to think the decision needs to be made up front in light of this.
However, there is a good deal of leeway that allows for an interpretation of the clause "when you or another creature makes a check/save" to mean "at any point during the process of making the check." In fact, that's the prevailing sentiment of this related question.
If we take this thought experiment into the real world, I don't think there's much to be gained by a DM denying an Artificer player the ability to use Flash of Genius towards the end of the ability check process. The reality is that most players will probably tend to use it that way anyway and enforcing its use at the start may increase the difficulty of play at the cost of increased frustration.