Aforementioned guide just relayed the information that at no time
during our stay are there any tickets left to see inside the
Colosseum.
This simply isn't true. Tickets for TOURS of the colosseum offered by the official website - specifically the underground - may be gone but general-entry tickets are always available and are not locked into any specific time or day. You can pre-order them here:
http://www.coopculture.it/en/the-colosseum.cfm
http://www.coopculture.it/en/colosseo-e-shop.cfm
Purchase adult tickets (€ 12.00 + € 2.00 reservation fee) and the € 2.00 reservation fee-only tickets for your children under 18. Choose the option to pick up all of your tickets onsite (at the ticket office reservation desk) as that's where you'll need to collect free tickets for your kids; you can't print those out online. Bring proof of age for your kids if any of yours are teenagers.
And do scroll down further on the page to "Individual guided tours"? Those are for the main part of the arena only and are only bookable by phone. If any of the days/times work for you and you want one of these tours, it wouldn't hurt to call and see what still may be open. Audioguides are available for rent as well (scroll down the page for reserving those). As you teach history, you might be able to get by with just prior research and a good guidebook; we purchased one at the bookshop there.
As previously stated, general-entry tickets are good until the end of December for any day, any time it's open, although if it's very busy the security check line may involve a bit of a wait, and also include entry to the Forum/Palatine.
http://www.coopculture.it/en/the-colosseum.cfm
Tours of the underground (mandatory: can't access on your own) offered by private companies are also available but usually cost quite a bit more. I think you and your children will be just fine with exploring the areas open under general-entry tickets, though.
Does this help?
Editing to add: your guide may be able to get in free as well or he/she can choose to stay outside while your family takes their look around.
List under "Free conditions":
...tour guides from the european union practising their professional activity;
Frankly, I'd question how good your guide is if they told you there's no way at all to access the interior unless it was one of the coopculture underground tours you'd specifically been asking about as those sell out within hours of being available. One other snag, though, would be if your chosen day was on the first Sunday of any month when Italian national museums are free: no tickets/reservations sold; colosseum to be avoided due to immense crowds and long, long queues.
The Colosseum is closed on May 1, December 25th and January 1st, and from 10 to 12 on Friday, 21th April 2017 so hopefully your chosen day isn't one of these. :O)