You do not need to reveal your salary!
Barring some regional or local law in your area (I have heard that Brazil requires you reveal your salary, but Brazil also has a lot more worker protections than most countries), then you have absolutely no obligation to reveal your current salary to a potential employer.
Company background checks do not uncover such information
When doing a background check, a company is not allowed to ask what a potential candidates previous salary was. A background check only reveals criminal history and verifies that you have worked at the places you claim on your resume and in the positions you claim.
The company refuses to interview you until you provide your current salary. A good number of companies operate on this policy. You are scared that you will not be hirable unless you comply.
Why are you leaving your current job? You are leaving your current job because you feel that you are being underpaid.
Why is walking away from this company a good idea?
Revealing your current salary to an employer basically removes your ability to negotiate salary with them. It is like playing poker, and revealing your hand to everybody at the table before you make a bet. That would not be a very successful poker player.
If the company is more concerned with your current salary than your skills and what you offer, then they don't really need or want you for your skills so much as they want warm bodies to fill seats for as cheaply as possible or they want people that they can easily control regardless of their contributions to the team. You could be the worlds best __ who ever lived, but to companies like this it just doesn't matter unless they know that they have the ability to manipulate and control you.
Advancement of career
If you wish to advance your career with a challenging and rewarding job, you need to ignore companies like this because they more than likely will not deliver the career growth that you may want. A company that is interested in you for your skills and experience will be more than willing to negotiate salary with you and pay what is fair market rates to have you on their team. Being able to negotiate salary also demonstrates confidence, strength, and these qualities make it more likely for you to be promoted for your natural leadership qualities in the future.
I applied to one well known corporation in my city and was asked to reveal my salary information on the application. I left it blank. Their HR called me and told me that I needed to tell them because it helps them filter out people who demand more than the going market rates. I said that I do not apply to companies that require this of me. They called me back the next day and asked me to come in for an interview anyway.
I went into the interview and it went very well. At the end the manager started negotiating salary with me and led with asking me how much I make currently. I didn't tell him and instead told him what I felt was a fair offer based on going market rates. Again he insisted that I won't be considered if I don't reveal. I immediately thanked him for his time and got up to leave.
They called me back the next day for the amount I suggested. I turned down the offer anyway because the environment wasn't a good fit for me.
It goes to show that if you are confident and have desired skills that you do not need to do this.