This could be done with stranded knitting because you are not carrying the color more than 5 stitches. If you want to do that be sure to keep the floats very long, do this by stretching out the stitches on the right hand needle just before you begin a new color. I hear you on the strands showing though.
You could do this intarsia and squares of intarsia are the easiest to do. You would use bits of yarn just long enough to do each square and little balls for the edges. I'm assuming you want to make the garter and the St st the same color and the seed stitch a contrast color. Whatever it works the same way.
You begin with the garter border at the end and then work the first edge garter stitches with it, then take the piece of yarn you want for the first block and start working with it. It will leave a little hole, but don't worry about that, knit the five stitches and drop that color and take another strand of yarn and begin working with it. You do that all the way across. Each color change will have its own ball or strand of yarn.
At the end of the row you turn and start back dropping and picking up colors as needed. Always pick up the new color from underneath the one you are dropping so that the yarns form an X, if you don't you will have a slit at the "join". You will have a lot of ends to work in, but if you really want to do this it is doable. I have to say I have never done intarsia with seed stitch, it may be slightly odd looking where the two colors join, but maybe not. Give it a try and see what happens.
There is a free video about basic intarsia on this site. I think it is under advanced techniques. I think there is one on stranding too (maybe called Fair Isle).