Background Check: Redundancy vs. Termination (Australia)

Hi all

Towards the end of 6 month probation at my previous role I got a tap on the shoulder and was asked to meet HR and the Head of Department. They told me they were not going to continue on after my probation due to not progressing at the speed they were hoping (underperformance). This was a surprise to me because I had only received positive feedback and thought I was going along well. After HR left the Head of Department stayed and said that between ourselves the reason I was being let go was cost reasons (workplace was very political). He also offered to be my reference and said that if things picked up again I could have my old job back (serious).

Anyway I've moved on and have completed an interview at another firm which is currently doing a background check on me. Throughout this whole time the story that I've been telling people is that I was made redundant (which is true). During the background check I was asked why did I leave my previous firm and I put down redundancy. However, some time later I remembered that HR at my old firm may have a different understanding of the story.

My questions are:
-can someone doing a background check ask about why you left (I'm guessing yes that's why the question was there)
-what should I do to not get my offer rescinded?

If it comes up I'm thinking I could ask my former Head of Department to vouch for me but honestly I don't trust him.

I'm hoping I could get the second most senior guy to vouch for me (he was my reference which was positive) but I don't know what else I can do?