How to split monthly profits in a co-owned (mortgaged) home when we invested different downpayments, and only one of us lives there?

You really need to wear two separate hats, the tenant hat and the (co)owner hat. Your brother will wear a co-owner hat as well. You should also, separately, loan your brother enough money that you make equal down payments. He should pay you back separately, outside of whatever you do with the house.

See my other answer here for more:Two siblings own home with mortgage. What is fair for both parties?

The way to do this is to separate the 3 different roles you're playing:

Role #1:

You and your brother as owners of the property who pay the mortgage. Who pays how much? How is that used to calculate how much of the house you 'own' when you come to sell it? (Ideally the answer would be 50:50, with an adjustment for the different down payments, so maybe more like 52:48).

What are you going to do if the rent doesn't cover the mortgage, or you have to find £10k for an unexpected expense? Who's paying for the insurance, and the utilities? These are all things you should figure out as if neither of you are going to live there. IE as if you were going to buy some other investment property.

Maybe you run it as a quasi-business where you keep a profit/loss account for the property (Rent, Mortgage, Upkeep) and pay out / pay in as necessary.

Then Role #2:

Your role as tenant. Which should just be to pay whatever the market rate for your room is, as if you were any other tenant.

And finally Role #3:

Your role as property manager. I would honestly just look at what property managers generally charge and pick something low-but-reasonable.

The key point is that you and your brother are still getting the same rent you always would have, and paying the same expenses you would have always had to. The fact that some of it is paid by/to you doesn't matter as long as the amounts are the same.

You can simplify the results when it comes to drawing up a contract, but you should explicitly agree the various different numbers between yourselves first.

Seriously, why not keep it clean and pay rent as if you were one of your own tenants? Yes, you get some of it back if you "pay" yourself out of rental proceeds, so why make it complicated?

Not only that, but there's a way to see you paying less rent as a theoretical loss, since what you pay is less than you'd receive from a regular tenant. Whether you pay less rent to begin with or pay full rent and get some portion of it back as "profit" is almost a distinction without difference in many ways. I can't speak to the tax implications of this - I'll leave that to much more expert voices here, but it's possible there could be some tax consequences either way which might factor into this, so you might also ask about this.

Is your brother okay with considering your work as "property manager" sufficient to offset the "lost" rental income a full-paying tenant would pay such that this is fair?

Perhaps one other solution would be for you to assume enough of the expenses of the property (i.e., some of the utilities, insurance, etc.) to make up the differential in what you're not paying in rent to balance it out.

While I can't speak to your family relationship, it would be terrible if at some point your brother were to come to a conclusion you're somehow "getting over" by paying less rent to live in a house he's helping to pay on.

My preference would be to keep it as neat, clean and simple as possible.