I’m sorry but … what an absolute mess
You mentioned that your main drain is a bit higher than the floor of the basement. I have the same situation, my drain is about two feet off the floor of my concrete basement. The washing machine drain was placed to drain towards the lowest point of the main drain. Everything else in the house with a drain is in upper floors.
I’d rearrange the plumbing to build a drain as close to the four foot level as possible for the washing machine … instead of trying to build a solution 6 or 7 feet up in the air … you’re asking for trouble if you are making your washing machine drain up to a high point.
I remember once renting a place that was an old house with a brand new concrete basement … the main drain was close to ground level that went to a septic tank and field. Their solution was to install the washing machine in the basement, drain down to the floor level, then to a grey water pump unit in a big box that was 3x3x3 feet … and that pushed the water up to the main drain. I don’t think washing machines are meant to push water that high up and you will probably constantly have a bit of grey water backing down to the washing machine.
When you do get that connection … go the extra step and install a ‘backwater prevention valve’ … it’s a giant one way valve that is attached to the main drain line as it leaves the house.
It allows water to always easily flow out but if it ever happens that water back flows into your house for some reason … the valve will stop any flow from entering your house.
I’ve always planned to get one but my situation is too difficult to install one.