Is it permissible for him to have intercourse with his wife in the bathroom?

My brother, may Allaah help you to do that which is good and make you satisfied with that which He has permitted you so that you will have no need of that which He has forbidden. What you have mentioned in your question may be answered by noting the following : 

1 – The modern kinds of washrooms in modern houses such as those in your country differ greatly from the places used in the past for relieving oneself, which were places where dirt, filth and impurities collected. Modern washrooms are not like that, rather they can be kept clean and there is no impurity in them. Therefore they are not like the places that people used for relieving themselves in the past, and there are obvious differences between them. On this basis there is no real reason why one should not fulfil one's sexual desire in such places when necessary, in the manner that you mentioned.

 2 – A man’s fulfilling his sexual desire with his wife may often happen as the result of stimulation, by looking or touching etc. Hence fulfilling his desire when it is provoked in such situations is a means of keeping himself chaste and lowering his gaze [from looking at other women]. This is what the Prophet (peace and blessings of Allaah be upon him) taught us to do, according to the hadeeth narrated by Muslim (1403) from Jaabir, who says that the Messenger of Allaah (peace and blessings of Allaah be upon him) saw a woman and he came to his wife Zaynab, whilst she was tanning a skin and preparing it for dyeing, and he fulfilled his need. Then he went out to his companions and said: “Woman comes and goes in the shape of a devil [i.e., she causes temptation], so if any one of you sees a woman, let him go and have intercourse with his wife, for that will take away what he feels in his heart.” This was also narrated by Ahmad, 19403; Ibn Maajah, 1853. Ibn Hibbaan narrated in his Saheeh, 4171, that ‘Abd-Allaah ibn Abi Awfa said: “The Messenger of Allaah (peace and blessings of Allaah be upon him) said: ‘No woman fulfils all her duties towards Allaah until she fulfils all her duties towards her husband. If he asks her [for his conjugal rights] even when she is on the back of a camel, she should respond to him.’” 

3 – At the same time, the Muslim should not forget, when his desire is provoked, to bear in mind the intention of keeping chaste and of enjoying that which is good and permissible. Then doing that will be an act of charity and righteousness on his part, as the Prophet (peace and blessings of Allaah be upon him) said: “In the intimacy of any one of you there is charity.” They said, “O Messenger of Allaah, if any one of us fulfils his desire will there be reward for him in that?” He said, “Do you not see that if he does it in a haraam way, that will be a sin on him? So if he does it in a halaal way, he will be rewarded for that.” (Narrated by Muslim, 1006, from the hadeeth of Abu Dharr). A man should also recite the du’aa’ that was narrated from the Prophet (peace and blessings of Allaah be upon him) for this situation. The Prophet (peace and blessings of Allaah be upon him) said: “When any one of you wants to have intercourse with his wife, let him say: ‘Bismillaah, Allaahumma jannibna al-Shaytaana wa jannib al-Shaytaana ma razaqtana (In the name of Allaah, O Allaah, keep the Shaytaan away from us and from that with which You may bless us)’, then if it is decreed that they should have a child from that, the Shaytaan will never be able to harm him.” (Narrated by al-Bukhaari, 6388; Muslim, 1434, from the hadeeth of ‘Abd-Allaah ibn ‘Abbaas).

 May Allaah grant you strength and bless you.