In Brussels, start with your French with your strong English accent, indicating that you are happy to speak either English or French (or indicate which is your prefered language.)
Those who prefer English over French will pick up your accent and switch languages.
What I have heard from Belgian people is that almost nobody speaks Dutch in Brussels, as a Dutch person with a clearly ‘northern half of the Netherlands’ accent I get a more friendly reaction. When they do speak Dutch, they will speak it with me, when not, they are happy to switch to English if they do speak that.
For someone with English as first language, English will be the first foreign option.
People who have grown up in the area should speak both French and Dutch and almost surely have learned some or much English. But a lot of the people who now live in Brussels grew up somewhere else, they hardly ever speak all three languages, (unless they grew up in Flanders or the country of Luxembourg) and often just speak French or just English if they are expats.
While Belgium people are supposed to always speak the right language, tourists get a lot more leeway.
When in Flanders, speaking with (older) people, offer the choice between English and French and some may take you up on your offer of French.
The older generations grew up in a time when French was taught in all schools but English only in some.
I even know some younger people who while fluent in both English and French prefer French if the person they talk with has it to an acceptable level.
French is the main language in officially bilingual (French-Dutch) Brussels, and as such the “correct” language to use there.
It is, however, entirely surrounded by Dutch-speaking areas, so if visiting the outer suburbs (such as Zaventem and Vilvoorde) you should use English there.
To clarify, Belgium has a French-speaking part, a Dutch-speaking part and a small German-speaking part, with Brussels being a French-speaking enclave. You do not want to use the “wrong” language, especially in Flanders, so when there, use English.
As you state, some Belgians are overly sensitive to language issues. Usually, as a tourist you should not have problems due to that, but of course you can always get unlucky and meet the exceptional sour grape.
Brussels is officially bilingual (Dutch-French), but in reality French is the dominant language. It is also an international city, so English (and a lot of other languages) are common. In parts of the town where you encounter a lot of tourists both languages should be equally fine. Both will be understood and a lot of languages are used. If you go to the less touristy parts, French will be your best bet. You simply are less sure that English will be understood and from your accent it will probably be rather clear that you are a native English speaker. Whoever you are talking to will probably switch if they think English would work better.
If you would travel to Flanders (the Dutch speaking northern half), English is your best bet. Simply because people, especially younger ones, are usually more fluent in English than in French. I fail to see who you are helping by speaking French.
In Wallonia (the southern French speaking half), I would start in French. Knowledge of English is rather bad, even though it is improving among the younger population. The same argument about switching to English as I described for Brussels is valid.
Source: I am Flemish (native Dutch speaking), working in Brussels. In Brussels I will often address unknown people in French. So I would definitely not be insulted if someone addressed me in French.
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