definitely yes re-do the washer drain.
Your two jpg drawings do not show the same thing; they don't "confirm" one another. As shown in the top view, you have four elbows for the washing machine drain before the drain water even gets past the toilet. Two 1/4 bends, two 1/8 bends. By redoing it as described above you get past the bathroom group and you only end up with two 1/8 bends. It can be done, you have said you can do it, and it needs doing, so I hope that is enough encouragement.
Your two drawings show some very sharp corners. The second jpg shows two 1/4 bends at or after the toilet that your first jpg doesn't show as 1/4 bends. Totally not understandable. You are so good at drawing, and then the drawings don't show the same thing.
In the second drawing, the cleanout is not pointed in the right direction, if you have a 1/4 bend there. Time to redraw or to re-do.
By redoing the washer drain, you'll be able to add another vent in one of the two walls that it crosses under. You can add another cleanout too. General remarks; I can't help more than that.
When pipes are a larger diameter, they carry the same volume of water over longer distances with the same "security" of characteristics, which enables calculations (about what will work) to remain valid over longer distances. The reason the Codes have been asking for ever larger diameters has a lot to do with that. In your cases distances are short, so the next most important limiting factor is venting, and with one more vent you will have gone over and above the minimum, and then (drumroll please) you won't have those physical constraints that caused Codes to require large diameters in the first place. Because distances are short (and number of elbows is reduced too, since elbows are equivalent to distances in their braking effect) and because venting is overprovided beyond the absolute minimum. Although that sounds like I know a lot and very precise knowledge too, I won't feel comfortable answering more than one followup question, worded loosely and generally. Also, don't quote all this as though it is to be picked apart and analyzed; learn on your own without asking me to explain more. (Obviously this applies to anyone else who posts here later too, asking for more). Sorry to be so much in disclaimer tone.
Are these pipes to be buried under a slab? I'd install more cleanouts.
david