An SSI recipient can own the house they live in and it is excluded from the $2000 resource limit. Qualifying for a loan is between you and the lender. Figuring out how to pay for the house and the upkeep on the house is your problem. However, if you get other money, it will affect the SSI.
I think everyone should talk to a banker or lender before they put an offer on any house to see if they can qualify and to know in advance how much the payments would be on any kind of loan and how much down payment is generally required. However, if a down payment of over $2000 is required, you would never be able to have that amount of savings and still be receiving SSI, so your goal may not be possible. You said you are close to $2000 now. The month after you go over the limit, you need to start returning SSI because you won't be eligible any longer. SSI will find out eventually.
You said something about using your boyfriend's income which would most likely mean you are co-owners of the house and both on the loan. Do you have a plan for what you would do if you and your boyfriend broke up? would you buy out his interest in the house or would he buy out yours? Or would you give it all to him or would he give it all to you? Marriages often don't last and a boyfriend relationship is even more fragile. So before you sign your name to a 10, 20, 30 year contract with a boyfriend, you need to have a plan for how to get out of it if your relationship doesn't last.
Also, if you decide to buy a house with a boyfriend, it is possible that SSI might consider you to be holding out as married to him (or not) and then his monthly income would be used in computing your SSI benefits (or not) which would likely reduce them. Can't tell you in advance what SSI would do. Not going to speculate either. It's just that the more you and a boyfriend act like a married couple by owning things together, the more likely it is that SSI will consider you holding out as married. So, as far as finances go, you need to either act like a married woman or act like a single woman. You will create problems with SSI if you are not clear.
How to compute countable earned income:
SSI counts the wages with the day of the paycheck, not the day that you did the work. Often a payday is in the month after the work was done. Start with monthly gross (before deductions) amount of wages. Subtract $65 for the earned income exclusion and $20 for the other income exclusion if you don't have any other income. Divide the remainder by 2. The answer is the countable income. Generally, income from one month determines SSI payments two months later, but there are some exceptions to that as well.
So, if you receive two paychecks in December for a total of $485 before deductions, SSI would be reduced by $200 in February - even if you were no longer working. The zero income in February would then determine the SSI payment for April. The zero income from October would determine the SSI payment for December. So, in the month you start work, you will have extra money to spend but two months later, the SSI will go down. So you need to plan and create a budget and not spend it all the month of receipt.
You absolutely are required to report buying a house and showing the documents and explaining where the money came from for the down payment. Also, when you live with others (boyfriends, strangers, parents), SSI also is required to know who pays what part of the household expenses to determine if someone else is paying part of your share of the food and shelter bills.