Any reason I can’t stack two sanitary tees here

No, there should not ever be two traps on a single drain line.

The point of the trap is to create a water barrier which prevents sewer gasses from coming up the empty pipe and into the house.

When you have two traps on the same line, you end up creating a kind of vapor block in the line which prevents it from draining properly.

What do you do?

Get yourself some 1 1/2" PVC (not CPVC, which is yellow, PVC which is white) with a 90 deg elbow, and also get two Fernco couplings (black plastic couplings with straps on each end that can be tightened) that will fit the pvc and iron drain pipe.

Use a metal blade on a sawz-all and cut out the p-trap in your basement. (Have a bucket handy!) Leave room to connect the Fernco coupling on both sides. I'd cut the right hand side out just past the coupling.

Measure and cut the PVC so that with the 90 elbow the ends line up with the newly cut drain lines. Glue the PVC together and confirm the fit, then connect to the old drains with the Fernco.

Viola - a proper drain line that will drain smoothly w/ no vapor lock.

What you are looking for is the Laundry-To-Landscape system: http://oasisdesign.net/greywater/laundry/

This system works very well and is easy to implement yourself at very low cost. I have implemented it myself and it works perfectly, providing much needed irrigation water to my parched desert yard (not a problem in Mississippi I imagine!). And unless you're washing poopy diapers, laundry is graywater, so it's not a health hazard. You could direct the water to mulched trees, or a gravel-filled trench you could dig. But don't just dump it onto the grass.