Will it come up that I was fired for gross misconduct?
DBS doesn't carry information about whether people were fired or not, what it carries about are criminal records of an individual, and few more things with the extended checks. As police was not involved (at least directly with you yet) there would likely be nothing to show on your DBS check. Except that you are in care of vulnerable people, which means you are likely staring down at Enhanced DBS check with barred list.
Problem is that this includes not just criminal records, but also something called "relevant police infomation" which can be a lot of things that relate to you and, what's likely what will sink you is the: barred lists check, and this is likely where what has happened may show up, as you can read:
How does the DBS decide whether to add someone to the barred lists?
As part of its decision-making process the DBS usually considers a range
of information from the police, as well as referrals from employers,
regulatory bodies and other agencies.
The DBS considers offences (convictions or cautions), evidence of
inappropriate behaviour and evidence of behaviour likely to harm a
child or vulnerable adult.
Before the DBS comes to a barring decision, the individual in question
is given the information the decision is based on and the opportunity
to explain their case.
This is likely what your former employee refers to, and this is likely what will sink your new job application. Even if you manage to get employed before those records get updated and manage to sneak through the cracks, this is not a happy ending, as applying for a job which you know you may be barred from is breaking the law, and eventually this will catch up with you as those check can and usually are redone at times.
So what should I do?
First verify what's on your record, you can do SAR to the DBS to find out if you are on the barred list, you may also do one with the police for good measure. And then probably start looking for a new job, just in case this one doesn't pan out.
If there is a record with DBS on you, you could try appealing it but as you did what you did, and wound up not getting into trouble with the police, I would personally just let it be, but whether it's worth taking the gamble and risking for the theft to be acted upon is for you to decide.