I know のみましょう but I was wondering if the above was legal japanese as well.
When in doubt, as someone Japanese, or else search for the exact word or phrase to see if it appears in Japanese blogs, forum discussions, online publications or other material.
There are a million hits for 飲んでましょう, in quotes, and 0.8 million more for 飲んでいましょう. So the evidence points to the answer that it is "valid" in the sense that it is in reasonably wide use by the people.
Reading some of these, it is difficult to see if there is any difference in meaning or distinguishing usage.
You would think that the progressive + volitional form would emphasize the ongoing activity. Let's do this, and keep going: let's be doing this.
This is making me curious.
A good pair to research may be 遊んで（い）ましょう vs. 遊びましょう. This would reveal if one is a preferred way to say "let's play regularly" versus "let's play now". I'm thinking that people would express this more often than "let's drink all the time". :)
Update from asking Japanese speaker, shedding some additional light.
This progressive volitional is appropriate if the activity is already going on. If people are already drinking (飲んでいる) then it makes for one of them to say もっと飲んでいましょう (let's drink more; let's keep drinking).
This applies to －たい (want to do).
飲んでましょう is a colloquial form for 飲んでいましょう. In general, abbreviating ～ている to ～てる is common in colloquial speech. Whether you count colloquial expressions as “legal” or not is up to you.
Now what is the difference between 飲んで(い)ましょう and 飲みましょう? As I understand it, the former implies that the suggested action of drinking is temporary.
I think that 飲んで(い)ましょう is typically used in the following context. A group of people decided to go out for a dinner (more precisely 飲み会, but I cannot explain it well in English…). Most of them have arrived at a restaurant, but not all of them. Then the organizer (幹事) says something like:
山田さんがまだ来てないけど、先に飲んでましょう。 Mr. Yamada is not here yet, but let’s start drinking.
Here the organizer says 先に飲んで(い)ましょう instead of 先に飲みましょう because the situation of 先に飲む (drink before something, here drink before Mr. Yamada comes) is supposed to be temporary, because people expect that Mr. Yamada will arrive soon.