What do I need to do so they must pay full replacement cost?
You're going to have to convince someone that $4,500 was a reasonable cost to incur.
Did you let the HOA know about this situation before making repairs? If so, when did you do that? If not, why not?
If you did notify the HOA in advance, did you receive a response? If so, what was the response? If not, did you follow up?
Did you provide the HOA with a quote for the work in advance of having it done? If not, why not? if so, what response did you receive?
When did you have the work done?
Has the HOA offered an explanation for wanting to pay only $1k? If so, what is it?
Since this issue was caused by a tree root, did you, in addition to replacing the concrete, take measures to rectify the underlying problem? If so, what measures did you take, and how much did that cost?