How does anyone know for sure that Oswald didn't have his hand gun at work that day?
They don't. The people who made up the narrative just decided that he picked up a handgun at the rooming house, even though there is no evidence of such.
I doubt that he did, but there is no conclusive way to know for certain. To get his handgun to the TSBD on Friday, Oswald would to have taken it to work with him on Thursday, hid it all day, smuggled it with him on his trip to Irving, hid it at the Paine home that night, taken it back with him on Friday morning.
Sort of like what he would have to have done with the alleged paper sack?
Oswald has no real need for a handgun that morning. He has his hands full pulling off the assassination. He has a loaded rifle as he escapes the 6th floor. If he needs to shoot anyone, he does so with his rifle.
And just hope that he didn't need more than one bullet.

Once he is off the floor, he is just another employee in the building. And in the end, we don't need to know why to accept his guilt in the assassination. The evidence proves he did it.
The evidence doesn't prove diddly-squat.