Mithrandir (another lord-of-the-ring-fan, I presume?)
While this is not my area of expertise and while I therefore can not give you a guarantee on the total corectness of my answer, a short google-research has given my the following information:
- you can simply go to the local authority who has issued you the Lohnsteuerkarte and ask for a second one. If anyone should ask you you could simply claim that you are having a second, parallel job. Thus at the end of the year you will have two Lohnsteuerkarten, please make sure you'll hand both to your tax adviser when (and if) you file for tax reimbursement etc.
- Ersatzlohnsteuerkarte is also possible, you would have to claim that the first was lost or destroyed. Costs a simple fee. But my guess is the first solution is easier and helps you to achieve your goal, hiding your huge salary increase from your new employer (congrats by the way and since this is my area of expertise: have you thought about starting some investment/savings plan with part of the higher monthly income? Had to ask...

)
With regards to the other part of the question: no, your new employer has not right to inquire about your former employment and salary and he has also no right to see any such specific information on the Lohnsteuerkarte
Cheerio