My take only!

A lump sum is a single payment of money, as opposed to a series of payments made over time (such as an annuity). It could be an agreement where an owner pays for a complete project by a contractor and does not require a breakdown of every cost.

1. I bought my house with one lump sum. Not as good as 3, but I bought my house by paying one lump sum would be equivalent to 3.
2. I bought my house in one lump sum. Not as good as 3, but next best. Compare I bought my house by means of a single payment (would be equivalent to 3).
3. I paid for my house in one lump sum. Best answer, compare "in one fell swoop", same kind of logic at work re that expression.
4. I paid for my house with one lump sum. Third-best answer, because in one lump sum is the better expression. Compare I paid for my house with dollars , which would be correct in terms of usage.
However, not one of 1 to 4 is grammatically wrong in my opinion.

My second question is can you use the expression "paid in full" to mean "one lump sum"? Am I using the expression correctly, in that context, in the following sentences?
5. I paid for my house in full. This could mean by 2 instalments, for example. So, it has a different meaning from "in one lump sum".
When I bought my house I paid in full. This could mean that I was unable to obtain any discount at the time of buying my house, ie I paid the full price for my house./i] Alternatively, I could have paid a deposit initially to secure the house, quickly followed by the balance owed, at which time I'd have paid in full, is another possible variant.
I think pay in full does not necessarily mean a single or a one-off entire payment - that's a problem, the ambiguity, that is.