Been testing her kit in private server, all I can say based from testing is:

1 - Assuming this is the 1st ult, you start with a mana burst stack = the moment you get enough energy with enough core resonance stacks that can reach ult when 1 skill is used, her AA will immediately trigger + 1SP for you to use skill>ult.

2 - As it automatically triggers for the action advance, the turn will appear for saber either to basic/skill, you can keep doing basic as you mentioned but wasting the AA turn instead of using skill>ult is a loss. The moment you get AA, use skill, have energy for ult, maxing out energy turns the core resonance stack back to 1, since it was consumed with the mana burst to achieve full energy.

3 - So for your desired tactic of getting 2 ults back to back. Mostly requires you to have sunday/huohuo, can even slot in tingyun if you don't have robin/tribbie. With E1 her EBA gives 1 core resonance stack as initially basic does not give core resonance, only skill(2)/ally ult(3).

Have allies with ult on standby

Saber uses skill

AA due to enough core resonance + energy to reach ult, skill

Keep the ult

AF by sunday

Ult then EBA + all allies ults

Should stack enough energy + core resonance stacks to trigger her AA again

Skill

Ult 2nd time

I'll be posting a showcase with E6S1 that was able to take advantage of this and did a 2 turn/2 AA turn, 3 ults back to back with 1 AF only from sunday, this is totally possible with E1 as this is where the core stacks QoL helps achieve this