Hello,
A lap joint requires cross cuts, but you won't find a cross cut blade with a flat top grind, only rip saw blades have flat top grinds.
Similarly, grooving blades are a rip cut, so won't do what you need. They are really too small a diameter for 10 inch saws anyway.
I would invest some time in making a decent router table, once you have it, it will be a boon for lots of other work as well as what you need for lap joints.
Otherwise,. a simple jig to guide a hand held plunge router is easy to make and is effective for lap joints and tenon cheeks too.
When I started woodworking, I thought a dado set or grooving blade would be useful for a TS, but, like everyone outside of America, found hat TS arbours are too short to accommodate them. 30 odd years later, I find I have never needed one and the is ALWAYS a better, safer and more efficient method of doing what I thought a dado set would do for me. I would learn to find alternative ways of making lap joints than the TS.
Mike.