Physio thinks I will need the manipulation and doesn't feel that resting will bring much improvement
Well, the physio would think that, because PTs are so number-obsessed.
Has he/she had a knee replacement, or is that just what he/she was taught?
It is far too soon to be thinking of having an MUA:
MUA: the ineffectiveness of early MUA (JBJS)
At the moment, your knee is prevented from bending because the swelling is making it stiff. It's like a garden hose that is full of water. The water makes it stiff and harder to bend. But empty out that water and the hose becomes much more flexible and will bend easily. Your knee is kept stiff by the swelling - and having an MUA, while it might force your knee to bend while you are unconscious, will only increase the stiffness.
In addition, it will cause new soft tissue damage, so your recovery could be set back.
Read this article:
MUA (Manipulation under Anaesthetic) and Adhesions
It's not exercising, or an MUA, that gets you your ROM - it's time. Time to recover, time for swelling and pain to settle, and time to heal. Your ROM is there right from the start, just waiting for all that to happen, so it can show itself.
There's no need to rush to get ROM (Range of Motion) because it can continue to improve for a year, or even much longer, after a knee replacement. There isn't any deadline you have to meet:
Myth busting: the "window of opportunity" in TKR