I have the same skin type as you and to be honest the fact that i would go brown after initially burning was my downfall.
I used a sun bed about 3 times in my life for the same reasons you did and also had the same reaction that you did.
I had two skin cancers (1 basal cell and 1 melanoma) by the time I was 30 and I've been told it is very likely I will get more. My dermatologist said that it was caused by sun damage and use of sun beds would definitely have contributed. He explained to me that it is very unlikely I'd have sustained the same level of burns in normal sunlight after the same amount of time of exposure, so sun beds are far more intense than actual sunbathing.
Those cancer scares freaked me out to the core. Then a friend of mine died of melanoma aged 35 leaving two young kids and she used sun beds; it was a fucking brutal death.
I loved and still love having a tan. Over the years I have tried out so many different fake tans with various different outcomes. These days I use Bondi Sands gradual tan (foam) and I rather love the fact that using fake tan now gives me control and is so easy to apply and the smell isn't that horrible biscuity smell.
If I get the Prime Minister gig next week 😜one of the first things I'll do is to ban those machines; they are close relatives of cigarettes in terms of health risks IMO.
I'm 50 years old now and long in the tooth. I've got some experiences under my belt now; amazing and fucking awful. Without doubt those two cancer diagnoses will sit with me as up there as easily among the most terrifying moments of my life. I still see a dermatologist annually and get a full mole check.
Do yourself (and the people that love you) a favour and please never, ever go on a sun bed again. And find a fake tan that works for you - it's out there somewhere.