Voter registration
Labour politicians may well be right that the Conservatives want to abolish compulsory registration because they (Conservatives) think that voluntary registration would give their party an advantage. However, it is equally true that Labour probably want to keep compulsory registration because they think that favours them.
It’s just like the argument about voting systems. Labour and Conservative politicians accuse the Liberal Democrats of wanting STV so there would be more Liberal Democrat MPs and Councillors but, of course, we all know that Labour and Conservative politicians want to keep FPTP because that gives them more seats nationally than STV would.
Politicians should no more decide such issues than generals should decide whether to go to war. An alternative for voter registration would be a period of public consultation (preferably with politicians keeping quiet, but that might be too much to expect) by the neutral Electoral Commission followed by a recommendation by the Commission.
Compulsory registration is wrong
I think that voluntary registration would be right for individual registration. Of course, compulsory registration is essential for household registration in case the “head of the household” decides not to register members of the household. However, with individual registration, everyone should have the right to choose for themselves whether to register, just as they have a right to choose whether to vote.
Compulsory registration should be kept only if household registration is kept, but then an alternative solution would be needed to the perceived problems of household registration.
On the other hand, the electoral register is also the basis for jury selection. Jury service is a civic duty that should be unavoidable except in certain, clearly defined, circumstances; it should not be avoidable by simply refusing to register. A possible solution may be to make registration compulsory for jury service with an option not to be registered for voting.