Interesting question. I'm probably the worst person to respond since brits are my weakest faction, but I play a lot of 1v1s with all factions and doesn't seem like anybody is responding. That being said, I also play against a lot of brits and can only think of very few brit's i've lost against in my more recent games (only lost to mencius and shadow since I've gotten "good" at the game). I think i've gotten to beating MOST brit players down to a science so I know what you mean by they are tough to play 1v1s with. Here are some general tips:
1. Don't move your HQ too early. It sounds like you know this already though. Making your source of reinforcement/unit building vulnerable too early can end the game early for you. Think about the HQ as your king piece in chess, you want to make sure it's safe. Make sure you've definitely secured an area before consider moving your HQ there. Every faction needs to retreat to their HQ to reinforce, so you're not "behind" if you haven't mobilized your HQ yet. Just because you CAN doesn't mean you SHOULD. Instead focus on controlling an important area of the map. Brit units tend to be stronger than their axis counter parts, so as long as your build order relatively matches theirs then you should be winning fights. This brings me to my next point:
2. Immediately contest a position of strategic importance. Brits do NOT build at the beginning of the game. So your advantage comes from being able to take a strategic area quicker than your axis counterpart. If the axis player sends their pios early without building, they've actually hurt themself badly because their pios are no where near as strong as your sappers AND they have greatly delayed their build order. From here, just keep taking advantage of your tech lead to win. More details about the opening in general, not as brits, can be found here: viewtopic.php?f=30&t=2575
Imagine you are playing chess in a 1v1 and brits is the "white side" who always gets to go first. You move first which means you set the pace of the game.
3. Build order wise, do you play low or high resources most the time? On low resources, there's a pretty standard build order almost everyone goes:
1 LT or Engineer (some players do AT squad first but i find the more successful players don't because they understand enemy build orders well enough that the first units available by axis will NOT be a vehicle)
2 AT Boys (around this time shwimms or sct cars are available by the time you make your second unit. if you are agianst PE, you might want a 2nd AT squad incase they scout car rush)
3 LT if you haven't gotten it already
4 Rifle squad or MG depending on your style
That's a pretty safe opening that should cover against 90% of the build orders that axis will throw at you.
If you play high resources, I recommend getting an LT first and dropping a glider right away so you have two sources of building units. Sure, it costs MP but it's not like you can SPEND that mp if you only have one building. More about my build orders here: viewtopic.php?f=30&t=1339
4. Most important, don't blame anything or anyone but yourself. The best way to get better is to find ways around the situation. You might be right that there are imbalances. Some units might be useless. Just don't use them. Use something else. For every complaint that brits suck there is an opposing complaint that brits are too strong. Find out why those people think brits are too strong and then do EXACTLY what they hate. I can go line by line through your blame list, but that argument won't help you.
Do I have to pick the RAF/Commando faction in order to stand a chance at all or can I still somehow win as Royal Engineers?
This really depends. IMO, the way to win as royal engineers is to understand macro. It comes down to overall strategy and decision making through the game. If you gain the upperhand in the early game and are ahead on tech, then go Royal Engineers because you can rush a churchill before the axis player can even field a unit that is capable of responding to the churchill. Then you just walk into the base and win. I've played tons of 1v1s as brits where i've had people just complain and complain about how OP churchills are and how they have NO solution for churchills... i've pointed out their errors but are still complaints for a reason. If you lose the early game, bouncing back as RE might be tougher. You may need to switch to a more defensive stance where you might choose to go the emplacement spam and arty howitzer route to keep axis at bay until you can field the comets to make cheap trades against the axis heavy tanks and finally closing with the churchill ace to slow push to the base. It all depends. In my opinion, I don't even choose a doctrine until I NEED to. So i won't go RE unless I have put myself in that "won the early game and will go churchills" situation. If i failed that, then I go something else. If i'm against a defensive turtle player I go royal arty. If i'm against an inf heavy player, I go RAF.