The spreadsheet is attached (bot_repair_eqs2.ods). However, I have to say, for
the biggest_bot_repair_value (that is the repair time of the Sawmill at max
damage) I used 241 in the diff, whereas the exact correct value would be 294.
You can easily adjust that in the spreadsheet in the cell
"biggest_bot_modifier". Obviously, the differences aren't that big. Please let
me know if you prefer the one or the other.

"The shear number of factors that we are using makes it very hard to evaluate
how well a set of equations works through playtesting alone"
I could make mathematical proofs, the formula will satisfy each constraint
listed above, if that would be necessary - maybe the constraint about the fun
could be a little bit difficult :D
However, the beauty of (multiplicative) factors is, that even an
non-mathematical user easily can see the changes from each input value without
even knowing the other inputs. That means, if you only now, that the
damage_amount afflicts the costs, but don't know any of the other factors, you
will easily see from the values ingame that double the damage will double the
costs. 
An other examples:
- If you increase your Repair_ability by one, all costs will be reduced by
approx. 2% - independently from the other inputs as bot types and damages (so
this is true not only for Acolytes, but also Sawmills and all in between and not
only if the bot has 1 damage or 299, but also in between).
etc. (I am sure, you've got the point :) ).

So, each input value can be evaluated on its own. That would not be possible, if
they were added/subtracted before multiplication.

Concerning the bot size:
In my proposal I used the max_HP for the bot size - as you intially proposed.
The healing_rate is NOT used for determing the bot_size, but only for the
computation of the healing the bot can make itself. Therefore, the bigger the
healing_rate the less costs the repair. Please also note, that the
difficulty_modifier is already included in Druidmap[bot->type].healing_friendly

I am totally ok with it, if you say, that the bot size should be computed by the
class or a some more sophisticated combination of bot attributes and not only
the max_HP. But for now, the max_HP are an easy and for the users easily
recognizable choice.
I would have no problem replacing it with a factor computed from the bot-class
or whatever. And it would be no problem at all for the formula, since you would
only have to change the one factor (bot_modifier) of the formula and it would
perfectly match into the overall formula without throwing up any problems with
the other factors. That's the second good thing about (multiplicative) factors. :)