Erm, abolishing the HRA would not stop the "abuse" you talk about. I don't blame you for getting this mixed up as the tabloid media deliberately conflates it regularly, but there are 2 key laws here, the HRA 1998 and the European Convention on Human Rights.
The UK has been a party to the ECHR since the 1950s. The ECHR contains all the substantive rights; right to life, freedom from torture or inhuman/degrading treatment, right to a fair trial and in these cases, right to privacy, and private and family life etc.
The HRA does not contain any substantive rights itself. All it does is make those rights in the ECHR enforceable in UK domestic courts. Previously, the applicant would have had to take their case straight to the ECHR to get a ruling, whereas now UK public bodies (including courts) have a duty to interpret UK law in line with the ECHR, unless the law is crystal clear in its intention to override ECHR law.
So all the HRA does is give further effect to the ECHR, even if we did repeal it it would only have procedural effect. We cannot just abolish the ECHR as international law doesn't work like that.
The right in question in the cases you're talking about is Article 8. Completely agree that this MP is being ridiculous, but if you google Article 8 you'll find it has 2 parts:
Article 8 ? Right to respect for private and family life
1. Everyone has the right to respect for his private and family life, his home and his correspondence.
2. There shall be no interference by a public authority with the exercise of this right except such as is in accordance with the law and is necessary in a democratic society in the interests of national security, public safety or the economic well-being of the country, for the prevention of disorder or crime, for the protection of health or morals, or for the protection of the rights and freedoms of others.
So there are two issues here, whether the dog engages Article 8 family life, which I think it probably does BUT there is only a violation if the not allowing the dog in can't be justified. I think there's probably an argument about protecting health by not allowing a pet in, so I'd be very surprised if he was successful when he took the case to court. I think he's probably just threatening to see if they cave...