Yes, it is possible to use mutual authentication with TSAPI/JTAPI.
On AES web manager, click the Security -> Host AA -> Service Settings menu. Now, select "Authenticate Client Cert with Trusted Certs" for TSAPI. Click Apply a couple of times and restart TSAPI.
On the client side, you will need to import your ID cert into the Java keystore used by JTAPI. The approximate command for this is:
keytool -v -importkeystore -srckeystore "myIDCert.p12" -srcstoretype PKS12 -destkeystore "jtapiKeystore.jks" -destkeystoretype JKS
If your ID cert uses a different CA to AES, you will also need to import your CA cert into the AES "CA Trusted Certificates" list.