You need to higher litigation counsel to represent you. I have dealt with Warner Chappell on many occasions, and it tends to be a honorable organization, but it is also very bureaucratic. You should find a lawyer (such as me) who has relationships with people in its legal department and can get the attention of the proper executives to deal with the problem. Chances are a demand letter will prompt an investigation within Warner Chappell, leading to relatively prompt action. Warner Chappell (I don't know much about Deep Purple) does not want to get sued because law suits for unpaid royalties are very expensive and, in all candor, in my experience many artists have grossly inflated imaginations concerning the amount of royalties which are due. In the event you cannot negotiate a settlement, you may need to sue (but note that you may have significant statute of limitations issues among others).
I recently represented the family of a very famous American songwriter of standard songs that most people instantly recognize---and the family could not believe that the quarterly checks of a few thousand dollars were appropriate. But when we looked at the paperwork, records of sales, and related documents, we discovered that the family was owed very little. People don't buy records the way they used to, and music revenues are, to say the last, disappointing. And the Performing Rights Organizations don't do a very good job tracking actual use of songs in smaller venues, resulting in a system that unfairly benefits well-established artists at the expense of everyone else.
Nonetheless, I know how to be aggressive in pursuing these matters with these companies when it is appropriate to do so. Before taking a case like this, a lawyer would need to sit down with you and review all relevant facts and details, including all applicable contracts, records of sales, payments, and many other details. Also, I won't take these cases on a contingency----I learned a long time ago that it does not work for either the clients or lawyers to work on a contingency fee basis on matters such as this. If you are owed "millions" are even "thousands" of dollars, then you would be wise to retain counsel to represent you----your investment in legal counsel may pay off in a big way if there are really "millions" involved. But if there are not "millions" at stake, legal counsel can give you a realistic assessment of the amount of royalties at issue and the best strategies for collecting them.
Record companies and music publishers structure contracts with artists and songwriters to obligate themselves as little as possible, putting the burden of chasing money squarely on those artists and songwriters, and without ever providing an attorney's fee clause to the prevailing party in litigation. That way, all they have to lose, besides their own legal fees which may just be their salaried legal staff, is what they actually owed you in the 1st place..
I think you know the answer to your question. You obviously need to "lawyer up," since your own efforts aren't succeeding and your claims aren't being taken seriously. The payors may also be "running out the click," since you've got statutes of limitation on any claims you have. Don't wait any longer to hire a lawyer and assert your rights.
Avvo doesn't pay us for these responses, and I'm not your lawyer just because I answer this question or respond to any follow-up comments. If you want to hire me, please contact me. Otherwise, please don't expect a further response. We need an actual written agreement to form an attorney-client relationship. I'm only licensed in CA and you shouldn't rely on this answer, since each state has different laws, each situation is fact specific, and it's impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.