well my reading of this is pretty clear - they are calling your bluff.. so call theirs.. even ask your solicitor to submit tribunal documents - you have nothing to lose if you've left. the month's notice is, i think, a red-herring ( you and yuor solicitor can easily position it as demonstrating how reasonable you are under considerable pressure and adversework conditions, quite reasonably it could have given them a good chunk of time to try to resolve the issues/have a proper exit interview etc).
Your solicitor has, wisely and reasonably given them clear cost warnings and a tribunal will respect that as 'being reasonable'.
Your employers are probably reading this tactically as an indication that you want money and perhaps think that they will use the (traditionally lower) tribunal predicted settlement to pressure you lower in a negotiation - so somehow give some gentle signals that you are seeking a public decision might help (ask youor solicitor if you think this has essentially become a negotiation or not)
They could win a tribunal case on constructive dismissal, but so could you, and they will probably be much more concerned about negative publicity and/or regular criticism at tribunal than you are, not to mention that a tribunal case will be more expensive for them than you as it will be HR time, legal time, press office time etc etc and costs are not awarded to either side.
Get your solicitor to ask the challenging and aggressive questions and even submit your statements, so that again you are seen to be being reasonable,(but just have a hard-ass solicitor) this will allow you and them wind-back space in the case of negotiation.
When negotiating, get a list of embarrassing procedural, policy and other screw-ups that will 'come out at tribunal and will be damaging' so that you can plkay these cards via solicitor when negotiating - up your settlement amount if they offer to settle on the doorstep at least by three steps of debate. (eg 'you said £45K, but I consider now that it will take three years to get to the same stage in my career, lets split at £60K) as they will be under considerable pressure if they do this.
Tell your solicitor that you feel sick and stressed in writing - this will help, and allow more adjournments if necessary.
Tribunals typically have one employer expert, one very experienced Union rep and a judge - who will question the employer harder than you.
Good luck!