[Landlord - US - California] Tenant sent demand letter

Hi all, first-time CA landlord here looking for advice. For background, the security deposit was $4,000.
I had tenants who originally signed a 12-month lease and were good tenants overall, aside from a few issues with parking in fire zones. We renewed their lease for another 12 months. About a month later, my wife’s parents’ health took a turn, so we decided to move back to CA to be near them. We began negotiating with our tenants to move out early, offering free rent and cash, which they agreed to. We then canceled the existing lease and had them sign a one-month lease ending July 31st.
During that month, they told us they’d signed a new lease elsewhere and would be moving out gradually throughout the month. To get access to the house sooner, we offered to pay for movers and help however we could. They ended up out by mid-month.

When they moved out, they didn’t request a walkthrough inspection. I asked if I could start having contractors in to begin work while the lease was still technically active, and they agreed. When the contractors arrived, they sent me photos for move-out documentation showing the carpet in terrible shape, a strong weed smell throughout the house, and trash left everywhere — no real cleaning had been done. We replaced the carpet, repainted the interior, and fixed several broken items while prepping the house for sale. It went on the market three days after we got the keys back.
We sent an itemized notice of the repair costs, showing they came to roughly $3,000 over the deposit amount. About a week later, they sent a demand letter insisting on the full deposit back, claiming we “pushed them out” — even though they’d already signed a new lease elsewhere.
I responded to their demand letter with essentially the same information from my original notice: that the damage went well beyond normal wear and tear, and that the carpet’s age was irrelevant since it had been professionally cleaned before move-in and wasn’t returned in comparable condition. Since I sent my itemized letter within the 21-day window, my understanding is I didn’t act in bad faith and shouldn’t be exposed to double or triple damages under California law.

What should my next steps be? Should I offer to return part of the deposit to head off court, or hold my position and wait to see if they actually file?