I purchased a Chamberlain KLIK1U 2 button universal garage door remote a little while back and remember there being a dip switch option. I'm not sure if you can program two dip switch openers to it, but it might be worth a try.
Another option which is the one I had to do is that if you have a button inside the garage that you press to open the door, you can get a pair of Chamberlain 955D receivers. It allows you to use a rolling code remote on the opener. If you used one for each opener, you could get one remote with several buttons and program it.
I hope this helps!
The emergency release cable on my garage door is inside the garage. When it is pulled, the door operates manually by lifting. If the springs are weak, it will take some effort to get the door moving.
You have two questions.
The first answer may be that your door has weak springs. A lever of some sort is indicated. Consider a two-by-four of some length, perhaps with a steel strap affixed to the end to get under the door. A fulcrum attached to the lever close to the end will give you some lifting to the door. Block the door at the height accomplished, then re-position the fulcrum, perhaps lengthening it to afford additional height. Block the door each time a gain is made. There are also devices known as fence post jacks which will allow straight vertical lifting once you have sufficient elevation to access the underside of the door. Even an old-style auto jack could be pressed into service, but I've not seen one of those in years.
At some point, you should be able to lift the door by hand, as the springs take over the load.
Rather than trashing the door should the above option be out of reach, consider to contact a garage door company for suggestions. With sufficient desperation, many other options are available. My neighbor has a bobcat type loader with fork-lift blades. If my door were stuck badly enough, it would certainly open under the force of the loader.
That leads my train of thought down another track. One could construct a simple set of wedges and tap them into place with a hammer. As the door began to lift, the wedges would be stacked onto additional spacers in order to continue the progress until the springs engaged.
If your release cable snapped and did not disengage the mechanism, it is necessary to perform the lift and remote actions together or risk damaging the opener. When my opener sheared the teeth on the motor, it was strong enough to prevent manual opening but not strong enough to lift the door. Had the release not operated properly, there would have been little to no additional damage to the already worn gear.
Best Answer
Likely there is not a learning remote for this, since they are different frequencies and protocols.
There are a couple of options, though:
Add a "Security+ 2.0" compatible receiver to the gate
Depending on the actual gate used, if you can wire in a universal receiver, then you can link the Liftmaster remote with it. Would need details on the gate control hardware to help with that (probably best to ask a separate question).
Of course, if "HOA" == "Home owner association" == you can't modify the gate, then this may not be an option. Maybe you could get the HOA to wire this in, as likely other residents would benefit because Liftmaster is a pretty popular brand (also sold as Chamberlain and many Sears brands, and probably others).
Custom-build a remote
Looks like the Linear remote is already a "keychain" remote. From there, get a normal Liftmaster visor remote, and try to fit the contents of the linear into it, or potentially get a Liftmaster keychain remote (smaller) and fit both of them into a visor-style clip.