I'll expand a little further on my thinking:
I can't tell from the picture, but those brackets you are using look like they have the room for 4 or 6 nails/screws/bolts, maybe 8.
and there are a couple of issues to consider. One, is that you'd ideally want the positive connections to have a bit of distance from the neutral axis of the structural members. You have 4X4 for post and braces so the neutral axis of each is 1.75" from surface. Draw those two lines in your mind. Then think about your fasteners and picture where the head and barrel of those fasteners are in relation to the neutral axis of each member. The closer they are to the neutral axis, the less effective they are
then, think about which direction the force of movement will be, and what you have to resist it. In this case it's the bracket, but physically, it's really the fasteners securing the bracket. And in the case of your application, the force of any flex will be parallel to the grain of the wood. Not only is parallel to grain resistance weaker, it's also more prone to accumulated movement weakening the wood around the fasteners.
that's why I suggested that 2X4. Yeah, the nails/screws/bolts you use in the brace will be moving, slightly, with the grain. But those fasteners you use in the post and the beam will be perpendicular to the grain. You'd think those would be at 45 degrees from perpendicular but that's not the case in terms of shear force. It will be close to perpendicular, and that's to the better. Whatever fasteners you use should penetrate thru the neutral axis. On those brackets of yours, you should have 3" fasteners. If you have 5" of wood using the 2X4 + 4X4 you should have 4 1/2" fasteners. Pre-drill about 1/2 the size of the fastener to keep from splitting the wood
if you use the 2X4's, keep in mind the neutral axis when fastening. Your fasteners should be closer to the edge of the members than the center. The drawback of course is that on the 2X4, flex will tend to weaken friction resistance over time, but that's counted in years, maybe a couple of decades. But that's still a shorter span of time than when the Blazers will win their next championship, although Olshey and Stotts may still have their jobs
Simpson Strong-Tie makes really good, engineered straps, but they would be pretty unsightly, and painting them fails pretty quickly, even with an oil base paint