Gaussian filter circuit
I've thought about this, and I don't think there's a commercially available single chip solution that comes close to doing what you need: in fact, you want to convert your rectangular function to a sequence of (mathematically) pulses, and then pulse-shape that with a Gaussian-shaped filter. Hard!

Now, the way I'd achieve this is:


use your very frequency-exact 10 MHz source to discipline the sampling clock of a fast DAC
Digitally generate the pulses


Now, how fast is a "fast" ADC? Ostensibly, it'll need to be significantly faster than 10 MHz; what you'll actually need depends on the "Gaussianness" of the pulse you'll need and the "width". You sadly couldn't put a value to that, so I'll have to leave that up to you, but let's say that with 10- to 20-fold higher sampling rate you can already achieve pulses that look pretty Gaussian if you add a sufficiently good reconstruction filter after your DAC.

Now, there's already a class of devices that are pretty much a fast DAC (and ADC), an externally disciplinable clock and reconstruction filters: SDRs.

So, judging from the frequencies and clock qualities necessary, an Ettus X300 with a basicTX daughterboard would give you (two baseband channels, IQ of) 200 MHz sample rate. It has an input for a 10 MHz reference clock. You can either attach a PC with 10 gigabit ethernet, and generate the desired waveform in software, or you can modify the FPGA image and generate it on the fly in the device. Disclaimer: I'm affiliated with Ettus/NI and these devices aren't cheap. Basically, they are nothing but programmable arbitrary waveform generators. If you happen to have other arbitrary waveform generators standing around, make sure they have a 10 MHz ref input, and use that.