Noterman, what you feel and how the bike stops are two totally separate things. What you felt wasn't you being pushed "deeper" into the bike but just a lack of stopping power. The truth of the matter is that when there is good traction the bike will stop in a shorter distance with the front only than with the rear only. This is due to weight transfer moving the weight of the rider and bike onto the front wheel and off the rear. In mild stops you can use both or either brake and not have any issues. But when you need to stop NOW to avoid a nasty then the front brake is your friend when riding on clean dry roads.

The oddball thing is that we mostly all stop with two brakes. But on a bike or motorcycle when you want to stop really hard you actually have to ease up on the pressure of the rear lever while pulling harder on the front in order to avoid skidding the rear. At the most extreme point of stopping ALL your braking is on the front wheel and the weight transfer is so complete that the rear wheel is barely in contact with the ground at all and has little or no traction. At that point you should not be using the rear at all, just the front. It's a fine line but this is where the bike and you will stop in the shortest possible distance. Nothing you can do will make it stop any faster. But like so many things using the brakes at this extreme is a skill. A skill that needs to be practiced frequently to learn it so it's there when needed.

As for going over the bars the thing to do is learn to modulate the pressure to avoid this. And it helps if you can lower your body and move your backside to the rear as much as possible to lower and move your center of gravity back. And if you feel the back coming up off the ground ease up on the lever a hair. Both moving your weight down and back as well as learning to feel for any lifting of the rear is part of that skill thing.

Now if you get into wet or loose surface conditions this changes. Often it changes radically. If it's really slippery then the rear brake is your friend since it will allow you to slide the rear a bit during a stop in a contolled manner where a sliding front will wash out and put you down pronto.

So like most things there are compromises and different solutions for different circumstances. And because of this I would agree with the others that say you should try to keep the braking separate by somehow arrangeing two levers that allow you a choice and then learn to use them really well by practicing maximum power stops before you actually need to do one. I also agree with Curtis about the issues with linked brakes. Regardless of what you do there would be only one situation where the ratio was right. The rest of the time it's a bad compromise.

You also haven't mentioned if this will be for a road bike or for a mountain bike setup. That would help with the type of suggestions. being offered. But either way I would suggest that you look at methods that allow you two levers so you can vary the pressure between them while applying them at the same time. This would provide you with the option of shifting the pressure for the type of surface and the stopping requirements of the situation. For a road bike that one described above with the two levers "toe to toe" sounded great. Then there's Curtis' overlapping system just above for a mountain bike. Another MTB option would be to use a TT style bullhorn lever mounted in the end hole and a regular one mounted so that the tips of the levers line up. Then by varying the pressure between the inner and outer lever you can vary the front to rear balance easily and naturally. Because you'd be using a road bike component in a mountain bike handlebar in this case some machining may be required.

With your strong hand doing all the braking I'd guess that it woudl be nice to move all the shifting over to the left side. Yes? At least that part won't be all that hard. Stacked rapidfire pods or a SRAM twist grip for one shifter and rapidfire for the other would make things tidy.

In any event whatever system you come up with I strongly advise that you set it up so that it favours the front brake as this is the brake you need to stop quickly in a shorter distance regardless of what you feel.