Second dog - introduction problems

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tl;dr
Will my female rescue get on with a younger male rescue given they have fought over a toy on the second meeting?

Current dog is a spayed female rescue - probably Boxer x Staffie.
She’s my first dog although I was brought up with a house full!
She's about four and a half and I've had her seven months.
Was in kennels for a few months, had probably been abused, was very traumatised and still nervous but hides it well.

Great dog now, good with people and kids, socialises with most other dogs.
As she has come out of her shell, she has demonstrated that she does not like female staffie sized dogs - if they get too close or snap she will snap back and not back down if they retaliate.
OK with male dogs that are not too forward and OK will smaller dogs.

She does not really play on her own, but likes running around with and stealing toys from other dogs. If she gets too excited, her hackles start to rise - then I stop her and we have a time out.
Walks out are off lead except for roads or if I see a dog on lead that she may not get on with.
In the home she snoozes unless someone (me) will play with her!

I thought she might like a companion so took her to meet a male rescue - similar sized, lighter build crossbreed. Looks like maybe some wire haired terrier and whippet / lurcher is in the mix along with many other things. About a year old, recently neutered.

First short walk at kennels - he got a bit forward, she snapped, he backed off then both walked OK
Second long walk out in the village.
Both on lead - fine
Mine off lead - fine
Getting treats and cuddles - both fine and no sign of food aggression or jealousy

However, in secure paddock and both off lead.......
Male rescue found a hefty stuffed plush toy and carried it proudly. He would not leave it on command but was happy to play a brief tug of war with me and no signs of upset.
Mine did her normal thing of not wanting to play with toys unless someone was already playing.
Male rescue then ran around a bit with it and I encouraged mine to run around with him - both seemed happy,
He then dropped the toy and I threw it - both chased it and mine got there first.
They started a vigorous tug of war which in a few seconds turned into a fight - I dragged mine off.
Male rescue took and then immediately dropped the toy and lay on his side for a while - very submissive - almost as if he'd been trained to do it?
Mine has small puncture wounds to her leg, he seems undamaged.
All this happened in well under five minutes in the paddock.

Given that I want a companion who will amuse her at home and who she can play with, can this behaviour be reasonably easily dealt with – or is this likely to be the wrong dog?
Or maybe, given that she does not back off if annoyed, am I best keeping her as a only dog?

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I have a male bullmastiff and a male border terrier who sometimes annoy each other.Both are neutered. They do sometimes clash, but as any kind of fight might result in the bullmastiff 'eating' the terrier, I have been very careful to not allow any behaviour that could develop into a fight situation.Your dogs competing over the toy is an example of where I would step in and take the toy from them, not allowing either to have it.They soon learnt the rules and are now best of friends most of the time !
As time passes I have developed a pretty good understanding of what to allow and what not too. I believe, based upon what I have learnt through my two, that most dogs will get along with each other if they have a set of consistently enforced boundaries.
My experience has been that introductions between dogs are an ongoing process that are as much about our behaviours as they are the dogs. My advice would be stick with it, be patient and most of all be consistent.If your experience is anything like mine the more you learn about your dogs the more enjoyable it is to have them around you.Mine are far from perfect and sometimes still misbehave,but that for me is what makes them individuals with their own personalities. I would go for it.