You do not have to use their approved shop in order for them to send an appraiser out. That's just flat-out untrue. Don't fall for it. I'm the same way with apps.

You can turn it into your company, as you described. If you do, ask them if they will waive your deductible, since the other company has already agreed that the accident is their fault. If your company will not do so, they will collect it back on your behalf when they collect their own money back from the other company.

Or, you you can get that adjuster's supervisor on the phone and tell them you are going to the shop of your choice, and that you will not be using the insurance company app, and that you will expect the other insurance company to set up your rental car and send an appraiser to the shop to look at your car, write the estimate, etc. Polite but firm will make this happen, even if you have to go beyond the supervisor level. If I were that manager and that call made it to me, I would A) take care of it for you immediately, and then B) ask the supervisor WTH they were doing letting that escalate to me in the first place, and then C) review the communication between you and the adjuster in order to give corrective feedback to that person.