Easiest way to regulate a 12 V battery to regulated 12 V?

What a regulator actually does is to smooth out variations in voltage to make a source look more like a battery. In a case like this, you don't have to worry about the source voltage changing much, so the main advantage of regulation is moot.

Often equipment that's designed for low-voltage DC will actually accept a range of voltages. Sometimes it isn't, and really does expect a regulated supply. The problem here is, what's the range that the synth will find acceptable? Admittedly, applying regulation solves that problem by nailing the voltage down to something that's supposed to work, but the thing is, voltage regulation gets complicated when the source and the output voltages are closer than about 2 volts, because regulators need some headroom to work with. Automotive batteries nominally sit around 13.8V; worryingly high if your device wants a regulated 12V input, annoyingly low if you want to regulate it to 12V.

If you can measure the voltage from your present AC/DC converter, while it's connected to the synth, with the synth turned on, and that voltage is with say, half a volt of the battery's open-circuit voltage, you would probably be able to directly connect the battery, without a regulator.

If you do this with a single battery, you will probably be fine. But if you decide to move it to a car, you must be very careful of load dumps. If you are using your synth in your car and you turn the headlamps off while the alternator is running, the energy in the alternator doesn't go away instantly. Instead, it appears as a high voltage spike - in some cases as high as 60V! I'd highly recommend a TVS and at least an LC filter to get rid of the noise which other systems in the car produce.

Also, don't forget an "idiot diode", to stop you reversing the polarity. Many manufacturers omit input polarity protection diodes.

Many musical keyboards don't particularly care about exactly what voltage they get, because they have internal regulation. Unfortunately, devices rarely specify the exact range of voltages they'll tolerate, in some measure because few wall bricks specify the exact range of voltages they might put out. A keyboard which nominally accepts 9-12 volts may be perfectly happy with anything from 6 to 15, especially if the voltage only goes up to 15 at times when the synth's current draw happens to be minimal, but if the manufacturer labeled the unit as accepting 6-15, someone might expect it to work with a so-called 15-volt wall brick that actually puts out 18.

Many devices with a 12-volt nominal input would be perfectly happy if driven directly by a car battery that was not connected a charger (or worse, something like a starter motor--starting a car can cause its +12 rail to swing up and down by dozens of volts).