彼は暑がりなのにやせ我慢をして真夏でもスーツ姿だって英語でなんて言うの？

"...no matter what" = regardless of.
"No matter what the government calls them, they are cuts"
We use the phrase ..no matter what' when we want to say that something will happen regardless of changes or what people may try to do.
"No matter what you say, I am going to leave you!".

You may use any of these expressions:-

1. He's unsuitably dressed for the hot weather.
   If someone is unsuitably dressed for the weather it means that what they are wearing is not the
   right attire for the weather.

2.The sun is scorching hot, but he insists on wearing this suit.
The word 'scorching" means very hot or blistering.

He takes pride in his appearance, he still chooses to wear a suit in the scorching heat.
- when someone takes pride in their appearance it means they always want to look good, look respectable and/or smart.
Scorching heat means really hot weather.

Even though he is sensitive to heat, he likes to wear a suit no matter the weather.
- this phrase explains that he stubborn, he still likes to look smart "no matter the weather".
This means if it's rain or shine he will always choose the suit.

Scotching hot means that it is very hot. This means that no matter what circumstances he wears his suit.

A) He always wears a suit come rain or sunshine.
*Come rain or sunshine-whether it rains or not; whatever the weather.
Example-"he runs six miles every morning, rain or shine"

B) No matter how warm it is outside he will always wear a suit.
Warm-of or at a fairly or comfortably high temperature.
"a warm evening"
Outside-situated on or near the exterior or external surface of something.
Example-"Anne put the outside lights on
*Wear-have (something) on one's body as clothing, decoration, or protection.
Example-"he was wearing a dark suit"

I hope this helps :-)

Ingrained is the perfect word to use when describing your friends habit (ingrained relates to a habit, belief, or attitude) it means his habit of wearing a suit is firmly fixed or established; it is difficult to change, even though it is detrimental (harmful) to him.

‘Now at nearly 21 the habit is too ingrained to break, he wears a suit even on the hottest of days'.

Persists is also another very good word to use to describe his action, as it indicates a course of action taken even in a difficult situation. the action is wearing a suit and the difficult situation is the heat e.g.
He persists in wearing a suit in hot weather even though it's detrimental to his health.

If you want to describe this person, you should feel free to use any of these explanations. I would prefer the first one because it has more details.