Transmitting a 350 MHz signal
You usually don't buy an oscillator running at your RF – for example, there's no single "2.4 GHz oscillator" component in the wifi module of your phone. (for the low frequency of 350 MHz, you can actually build a relatively bad oscillator using a SAW filter and a transistor yourself, but don't.)

Instead, you buy some oscillator that generates a stable base oscillation, and then use a frequency synthesizer that generates a much higher frequency from that base oscillation. In the simplest case, you'd just buy a PLL chip, and a crystal oscillator running at an integer fraction of your target frequency (350 MHz).

A very much hacky way of getting a multiple of a base frequency is getting an output buffer that generates a square wave from the base oscillation, and then filter out the fundamental and all the harmonics you don't want, leaving you with just the harmonic you want. This works through the the discrete nature of the Fourier transform of the square wave. Look for the Fourier series representation of the square wave to understand where the harmonics appear!

Most RF SoCs include much more flexible designs for generation of clocks (see: fractional-N synthesis), where a much larger range of frequencies can be selectively synthesized from a reference oscillator.

The honest answer to how to build such a device is hence probably:

Look through the RF SoCs (which typically are a microcontroller, ADC/DAC and an RF frontend) of the major semiconductor producers. For example, Silabs has quite a few RF/microcontroller combos (example). That's the style of component most companies use when they need to add a sub-GHz RF interface to their devices. The example I picked can produce a whole range of frequencies, nearly continuous from 110 MHz to 965 MHz. It's not 100% an SDR, but chances are that the actual comms standard used by your remote control is indeed covered. You will, however, need to add impedance matching, antenna or connectors etc yourself. This is just a chip! Maybe there's cheap eval boards, I haven't looked into that.

Other than that: When paying for the HackRF (or other SDRs), you're paying the price for getting a device developed that works on a very large range of frequencies well, is tested, blocks out frequencies you're not interested in producing or receiving, comes with a whole firmware and driver infrastructure; when you don't need that level of sophistication, for example because you really only want to use a very limited set of frequencies and standards, things get cheaper, but you'll have to do them yourself, because they are now application-specific engineering of RF hardware.