Short answer:
No.
Long answer:
Even more no. The Underhanded rogue talent has no interaction with the flat-footed condition, only the surprise round.
There are a couple of ways to make an opponent flat-footed during combat, but none of these would allow you to treat the opponent as being in the surprise round.
Invisibility would not affect Underhanded in any way, since:
a) It doesn't change the fact that you're not in the surprise round.
b) A concealed weapon refers to the Sleight of Hand's "Hide Object" action, not just any weapon your opponent cannot see.
Also, your opponent is only losing their Dexterity-to-ac if you're invisible, they're not becoming flat-footed.
The tricky part about Underhanded:
Drawing a concealed/hidden weapon is a standard action. Unless you possess some way to reduce this action or circumvent it, you can't normally draw your concealed weapon and attack in the surprise round.
The Deft Palm Rogue Talent (Ultimate Combat) allows you to conceal a weapon in plain sight, which circumvents the need for drawing the weapon.
The Sandals of Quick Reaction wondrous item (Ultimate Equipment) would give you a both a Standard and Move action during the surprise round (although you can't combine them into a full-round action). Together with the Quickdraw feat, this would allow you to draw a concealed weapon as a Move action and then attack with your Standard action.
Spring-Loaded Wrist Sheath ( Adventurer's Armory) is by far the most elegant solution. For 5 GP you can hide/conceal a weapon in this wrist sheath, and release it as a swift action.
Actually, you can be concealing the weapon and have them not notice it. Hold it behind your back? In any case, if the weapon is concealed at the start of the surprise round, it is unlikely that you can make use of underhanded.
Underhanded explicitly states it is to be used in the surprise round, no matter how much rationalizing you want to say that it is talking about the flatfooted condition.
A rogue with this talent gains a +4 circumstance bonus on all Sleight of Hand checks made to conceal a weapon. Furthermore, if she makes a sneak attack during the surprise round using a concealed weapon that her opponent didn't know about, she does not have to roll sneak attack damage, and the sneak attack deals maximum damage. A rogue can only use the underhanded talent a number of times per day equal to her Charisma modifier (minimum 0).
Flatfooted isn't even in the description of the ability, so I don't see how it is relevant. You could use Underhanded against a target that isn't flatfooted (such as with Uncanny Dodge).
Here are some alternatives to Surprise Strike. All found in the span of 30 seconds with the Archives of Nethys search function.
Catch Off-Guard (Bear in mind that creatures with Natural Attacks are considered Armed)