TL;DR: (1) we don't need to go very far to measure the spectrum of Earth's reflected light: a satellite in orbit around the Earth could easily do that; however, (2) detecting the reflected light of an Earth-size planet is extremely difficult: we currently use other techniques with much greater success.
The Wikipedia article on how scientists detect exoplanets provides a good introduction and overview - well worth a quick read. In summary:
Most of the thousands of exoplanets detected so far have been found using the transit photometry method, where the measured brightness of the star temporarily dips because a planet is blocking some of the light as it transits across the face of the star.
The next most successful technique is to measure changes in the radial velocity of the star to deduce the presence of a planet (by the "wobble" its gravity causes). This was the most productive method until 2012, after which transit photometry became the dominant detection method.
However, neither of these methods are looking for reflected light from the planet.
It is possible to deduce the presence of a planet from periodic changes to a star's brightness that would correspond with a nearby planet going through its phases (like how the Moon is bright when full and "dark" when new), but we're typically talking about a Jupiter-sized exoplanet in a close orbit - definitely not a relatively "tiny" Earth-like planet further out from the star.
It's also possible to directly detect the reflected light, or the infrared light emitted by a "hot" planet, or light that has been polarised by the planet's atmosphere – but all these methods rely on a much-bigger-than-Jupiter planet orbiting a relatively small/dim star, and preferably in our local stellar neighbourhood, since the light we're trying to detect is exceedingly faint. Note that currently the smallest directly-imaged exoplanet is still more massive than Jupiter, and Jupiter itself is over 300 times more massive than the Earth (and ten times the Earth's radius, which is even more relevant).
Note also that Proxima b (an Earth-size planet orbiting our very closest neighbour, the red dwarf Proxima Centauri) was discovered using the radial velocity method, and while efforts are under way to observe it directly, no direct image has yet been obtained. This might give you an indication of just how hard it is to directly image such a (relatively) tiny planet!
Also, we don't need to send a probe to the edge of the solar system to work out what we're looking for, as we already know exactly what the reflected light from Earth looks like – the iconic Earthrise being the most famous example. And in 1990 Voyager 1 took a famous photo of the Earth from 6 billion kilometres (41 AU) away: see the Pale Blue Dot.
By Sept 2017, Voyager 1 was 21 billion km (140 AU) from the Sun - by far the most distant probe we've ever sent. Nonetheless, that's a long way from the "border of the solar system". Voyager has entered the interstellar medium but it hasn't yet left the Kuiper Belt, and it will be another 300 years before it reaches the Oort Cloud, and tens of thousands of years before it reaches the real edge of the Solar System.