Since your house is on a slab your "Main cleanout" should be located on the main drain about 3" outside the foundation wall.
You will not be permitted to tie in at the main cleanout because the main cleanout is a rodding tee and may only be used for cleanouts. (Rodding Tee's have a bi-directional sweep which permits a drain snake to be run in either direction.)
Being in south Florida and having a septic tank, the main drain is probably not much more than 12 to 18" below grade.
The is a code restriction that prohibits digging a trench parallel to the foundation wall within a 45 deg downslope from the wall. The is to prevent disturbing the soil which could effect the foundation. Rule of thumb, when running a trench parallel to the foundation wall you must be one foot away from the foundation for each foot of depth. You may dig a trench to the wall at right angles.
Usually the soil in Florida is very sandy and loose so you should be able to locate the drain line with a probe before you begin digging.
I would run the line horizontal from the washer location through the wall in the manner you suggest. Immediately outside the wall install a sani-Tee and drop the line vertically below grade. Install a long radius 90 at the bottom of the vertical drop and run the line to the house drain line. Tie into the house drain line with a WYE.
On the washer end you will need to use a Wye and 1/8th bend or a Combo to make the vertical riser for the vent. (May not use a sanitary Tee)
The code allows you to reduce the diameter of an auxillary vent to 1/2 the diameter of the line it serves once it is 6" higher than the flood level rim of the highest fixture served (in this case the top of the washer standpipe), but in no case may it be less than 1-1/4".
You might be able to find a 2" air admittance valve at a plumbing supply store, but the easier method would be to extend the vent riser 6" higher than the top of the standpipe then install a 2" couling with an 1-1/2" x 2" bushing and an 1-1/2" male threaded cleanout adapter. (That will provide an 1-1/2 female NPT thread to accept an 1-1/2 AAV.
The following illustration will better explain when to use a Sanitary Tee of a Wye & 1/8th bend.