Goose would NOT prop Tony Stark
From the clips I've seen, Goose has much better taste than to prop a self-centered, misogynistic war profiteer.
Note: I'm not sure what happened to Goose, so in this story she's visiting the compound because she's Goose.
***
Tony didn't visit the compound often. So he was shocked to walk into the common room to find an orange tabby cat on Cap's lap. The cat was leaned against Cap's chest, purring as Cap slowly stroked down the long line of the cat's back.
He crossed his arms and said, "I don't remember anyone asking permission to get a pet."
Cap kept his focus on the cat and said, "Goose isn't a resident. She's a visitor."
Tony walked over to the couch and said, "Doesn't make a difference--my place, my rules. And I never made a rule welcoming four-legged occupants."
Natasha walked over from the kitchen area and said, "You didn't make a rule against them, either."
She sat and offered a hand for Goose to sniff. Goose lowered her head so Natasha could scritch behind Goose's ears. She shrugged and said, "Goose is a special case."
Tony frowned and said, "I want her out of..."
His voice trailed off as Goose slowly turned her head and STARED at him. He felt like Goose knew everything he had ever done--every misstep and every crime.
She was judging him. He didn't think he passed.
Instead of his original plan to lean in and either pet the cat or pick it up and chuck it out the door, Tony backed away and said, "Never mind."