Bù kèqi or Bú kèqi: In PinYin, do I write the fourth or second tone with 不客气?
That depends on why you are writing it. The standard, normal way of writing it is to disregard the changed tone and write the original tone, i.e.  bù kèqi. The same goes for tone changes of 一 (yī). Not showing the tone change is not a problem in this context, because the rules governing how to change the tones are quite simple. Since most Pinyin is written for native speakers, there's no need to include something which is obvious to the reader.
However, if your goal is to highlight this tone change, such as when teaching someone the language (someone for whom the tone changes are not obvious), it can sometimes make sense to write the actual tone for 不 and 一, so bú kèqi in your case, and yí yàng for 一样 or yì qǐ for 一起.
The third tone is (almost) always written as a third tone, though, as it would otherwise be impossible to know if a second tone is actually a second tone or a third tone that has changed. The only place where níhǎo would be acceptable for 你好 is in a discussion about third tone changes.
The problem of lost information does not apply to 不 and 一, so in teaching materials, changing the tone is usually a good idea (nothing lost; something gained). So, which one you choose depends on why you are writing it: If it's to teach someone pronunciation, then change the tones; if it's for something else, then leave the original tone!