Starting 9->10 upgrades, which is most effective strategy?

I'd combine some of the strategies you mention.

If you know you're not using one bot in particular leave it for last.

Between the bots you know you'll use, compare relative improvement vs each-other. One metric is rate of improvement per day of upgrade time. If you level the Galahad from 10 to 11 your weapons improve by 10% while your hitpoints improve by 6%. Because the hitpoint improvement means you can use your weapons ~6% longer these compound, meaning the bot got 16.6% better at doing damage. It takes 6.9 days to do the L9 > L10 upgrade, so this works out to the Galahad getting 2.4% more capable per day.

Compare this to the Lancelot: it only gets 1.5% more capable per day L9 > L10, while the Griffin is a little better at 1.8%. Judged this way upgrading the Galahad is the value, and upgrading the bot before the weapons is slightly better than the other way around.

Repeat for L10 > L11 and the Galahad is 2% per day, L11 > L12 is 1.6% per day (still better than the Lancelot L9 > L10).

Extend this analysis and assuming you're limited by time (and not silver) the ideal upgrade order is: Galahad > L10 > L11, Griffin > L10, Galahad > L12, Lance > L10, Griffin > L11, Lance > L11, Griffin > L12, Lance > L12

If you're silver constrained you'll need to run a different analysis for that. I'd also move things around to prioritize for a "favorite" bot you know you do well with, but use the above concept as a general framework.

In general this will resolve for keeping the bot equal to or one level higher than the weapons (some exceptions for bots that gain hitpoints quickly). It will also favor upgrading lighter bots and those with fewer weapons ahead of heavies with lots of weapons.

$.02

WoW great statistcal analysis comparing dam vs wepaons!!  Really like thoughts on compounding the damage very in depth and analytical.

Favorite bot would be Aphid/Orkan Griff.  But the Galahad is work-horse bot with best effectiveness for time spent.  Will start there.
Silver is still not an issue, still close to 337 Million Ag.  If I keep Dam around 450k +, I will good for a while.  I will do similar analysis for Ag, to see where I will start to run out to make decision on purchasing Prem package.
One downside with the current battles is the continual frustration has me playing less overall, diminishing my Ag bank.  The cost/time has not diminished my Silver bank in any perceptable amount, but it will soon.  I hope my desire and play time will both increase to offset the additional cost.

Thanks again for the great feedback!

I'd level the fattest, flat out brawlers first, then hit and run bots. Since they've levelized all the bot (and weapon) progressions now, you would get the most incremental benefit doing the fattest bots first. Same with weapons.

Under the old scheme, I did 9/9 >> 10/10 >> 10/12 >> 12/12 (maxing still in progress). Seemed to work pretty good. At 10/12, I could compete ok against max bots except 1-on-1 where you could tell the difference. 9 bots were brutal against max hangers.

But silver is your constraint, so plan accordingly.  Especially if getting premium is required to finish it off.  Getting premium earlier I think is better than later, because if you have the nice silver bankroll, you can spend gold to accelerate the last levels, whereas if you are too silver constrained near the end, the gold is sorta useless and just sits there, mocking you.

Not yet.

That's one sad looking Rhino...almost as sad as the HydraSpriralRhino™ I have seen recently in Springfield

As for the upgrade - I'm still below L9, but as far as L8 is going, I've noticed more balanced play since I got my bots up in upgrades.

Haha.... I agree, that was for illustration purposes only....  
Not sure what my 5th TT bot will be? - Rotate the slot
*Gareth / Leo / Carnage / Rhino / Natty