Hi Mark,
I'm quite happy with "at once" in that sentence too. If anything it's the "I was able" which strikes me as less likely than "I managed to" - but I suspect what's happening with both of us is that we're just reacting against a phrase that we personally don't use very much, preferring an alternative - but which other people do use. The sentence as it stands is really fine.
Would you, I wonder, tend to use "at once" more with the meaning "at the same time" as in these titles I found on Google:
Many vaccines at once OK for kid's brains / Why three buses come at once and how to avoid it. Interestingly, if you Google it, it's this meaning that predominates - I didn't come across an "immediately" meaning until the second page, where I found
End Gaza's Humanitarian Crisis At Once and the next example didn't come up until page 7. The other expression that came up frequently was "all at once" meaning
suddenly - eg
All at once he realised what she meant.
After a long search, I finally came up with these examples, all from American sites (all spelling mistakes, lack of capitalisation etc were in the originals) :
i hate how long it takes but i never do it at once. and either way you dye it, your still gonna damage your hair the same amount.
If you can't do it at once, great… than enjoy the process of learning, falling out, going back in and eventually suddenly getting it! 
...
I tried it at once and I am very, very pleased!!
I did it at once but using that many coupons at Whole Foods is my max.
It's ireversable? I did it at once, but I can't remember HOW I did it. I'll mess around and try and remember
These examples would seem to contradict Breadbaker's idea that the expression is only used in formal contexts in American English, as they all contain other elements that clearly mark them as informal. My intuition about British English is that it's a neutral expression - ie it can be combined with elements marking either formal and informal language without sounding out of place. In the dictionaries I checked, it's not marked for formality at all.