KSA medical, Visa agent vs done elsewhere

im going from UK to saudi.
I need to get my visa medical done and so does my wife who will be going with me.
Visa agent based in london wants me to do it 3 hours away in london. We will have a new baby soon, so its impractical to get it done so far away and expensive.

Will it make a difference if I get it done at a private clinic locally? The visa agent is being coy about it being accepted.

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