Hey, welcome back 
Any transition path that we want to solve has to end in a stationary general equilibrium (but does not have to start from one, although most papers do so). Intuitively, if the end of the transition path is not a stationary general eqm, why would you end up there? [Actual analytical theory proving this is sparse at best, there is little to no economic theory on existence, uniqueness, etc., for transition paths. Is more just informal and intuitive standards.]
Because rising debt is not possible in a stationary general eqm, you cannot have rising debt in the final eqm.
You can however still have rising debt over the transition path, as long as at some point it comes back under control and settles into a constant level; note that you could put this final eqm 300 years off into the future so that for the next 100 years the debt just goes up. [There is probably some thing that can be done to marry it with a model of sovereign default; maybe that is now in the sovereign default literature, but it was not really being done last time I looked.]
So how can we set up a model with government debt that rises for 100 years, and then settles into a level? The first thing that comes to my mind would be to use a fiscal rule, and then 100 years into the transition path change the fiscal rule. The easiest kind of fiscal rules for transition paths are those that adjust spending and/or tax in response to deviations of GovDebt/GDP from a GovDebt/GDP target. The advantage of this kind of fiscal rule is that you know in advance what GovDebt/GDP will be in the final stationary eqm (it will hit the target). This avoids having to jointly solve the transition path with the final stationary eqm (which is what you have to do if you don’t know where you will end up), allowing you to focus on solving the transition path.
Your first 100 years could then just contain whatever fiscal rule you want, e.g., just keep spending/GDP and revenue/GDP constant, and accumulate any gov budget deficits into the ever rising Gov Debt.
The switch from the ‘unsustainable’ fiscal rule to the ‘sustainable’ fiscal rule might be rather abrubt, so you could alternative introduce the ‘sustainable’ fiscal rule in period 100 with a coefficient of zero on the GovDebt/GDP deviations from target, and then gradually increase this coefficient over the following periods. Notice that the more gradual this is, and the further into the future it occurs, the less impact it has on the first 100 periods of the transition path; so this allows you to decrease the importance of the assumptions you make about it to the results that you care about in the first 100 periods.
PS. What you describe, where you have an ‘unsustainable’ fiscal rule, which switches into a ‘sustainable’ fiscal rule when GovDebt/GDP reaches some threshold should also work (you are doing the same thing I describe, but the switch in your case occurs in a period you don’t know in advance because it is endogenously determined). I would suggest seeing if you can first solve just the ‘second half’ of your transition [so start the economy at the GovDebt/GDP that is your trigger for switching, and see if you can solve the transition that uses the ‘sustainable’ fiscal rule. Remember you don’t have to start from a stationary eqm, so you can just take your regular economy and crack the debt up, start from there]. Once you can solve this ‘second half’ correctly you could then try returning to solving the whole transition (maybe the switch is also very abrupt, so you can phase it in like I described; this is probably not necessary, but might be a smoother shape and so easier to solve for).
PPS. Fiscal rule means that your model has the Government budget,
G_t+(1+r)B_{t-1}=R_t+B_t
where G is gov spending, R is revenue, B_t is the amount the government borrows this period. And then alongside this you add a rule for spending (or tax rates, or both) like
G_t=G^*-\phi (B_t/Y_t-BdivY^*)
where G^* is ‘typical’ spending, and then if the gov-debt-to-GDP-ratio, B_t/Y_t is higher than target BdivY^*, then you reduce spending [and/or increase taxes]. One advantage of just solving the second half of your transition path after you switch to the ‘sustainable’ fiscal rule is you can figure out some sensible parameters for the fiscal rule (\phi). Alternatively you might use a rule based around both debt and gov spending to GDP ratios,
G_t=(G*/Y_t-\phi (B_t/Y_t-BdivY^*))Y_t
[I have just modified it to be a rule in terms of G_t/Y_t instead of G_t. The two will only differ in practice if you have big changes in Y_t, in which case the later might be a better choice.]
