It could be worse. You're not asked to price rate exotics like accreters that might need more inputs besides implied vol cube :) and you're only asked to make markets. I.e., if I understand the question correctly, you don't intend to keep much rate vega / sensitivity to the poorly observable implied rate vol, unless in your view it's very mispriced.
For a market maker, it's a good practice to publish daily the indication-only bid and offer of your volatility cube, and market commentary attributing the market moves; and update intraday in case of large market moves. Your customers should be able to calculate their own mark to market, rather than accept yours as gospel. A customer would be happy to be able to plug the same interest rates and implied vols that you use into e.g. Bloomberg pricer and get MTM similar to yours. You should treat your customers equally by publishing the same indicative quotes for everyone. Giving different indicative quotes to different customers leads to problems.
You are unlikely to need an overly complicated pricing model that supports negative rates for EM.
Your customers should know your inducative vol quotes, to know approximately what it would cost if they decide to unwind early, without the need to contact you. But assuming that they will mostly trade one side, you should find hedge funds or reinsurers that will take from you the vega that you should not retain. You should not start trading until this part of the puzzle is lined up.
In addition to the trades that actually print, your bid and ask will be affected by your axes / being a better buyer / seller, i.e. having unwanted vega that you want to flatten, or a view that the vol is mispriced now.
You will also anticipate / take views on customer activity. You don't necessarily need a fancy model for that either, but you will learn that some customer always buys or sells some vega evry month, and another is always willing to trade your unwanted exposures at fire sale prices. Avoid letting such considerations spoil the markets for customers who recently traded with you.
This means that if a customer buys something, and you say the next day that it now be bought for less, then the customer becomes unhappy. To avoid this, avoid marking things cheaper for a short while after they are bought. Or you can let the customer get an adjustment/refund, but this is more work.
Indicative only means that if someone does want to trade, you must check again that your numbers still make sense. No one can hit / lift / force you to trade if you no longer like your own quotes. You can also treat some customers better at this point.
But where do you get the implied vol cube in the first place, before there are trades? I would approach this problem as follows. There are only on the order of 10 emerging markets currencies with liquid rates vol trading. I would collect all imaginable explanatory variables available - onshore swap rates, treasury rates, cross-currency bases, fx rates, sovereign CDS spreads and quanto factors, prices of key commodities, equity indices - and their bid-ask spreads, historical vols and implied vols if available :), and just run lots of linear regressions to see what might be driving their rates implied vols unfer normal market conditions, not expecting much correlation to most of these; and guessing they might get correlated when the markets move a lot. Then make up the rates implied vols for the new market. Let your fantasy run wild, but start with very wide bid-offer spreads. :)
In addition to caps and floors, you may want to consider options whose underlying would be exchange-traded interest rate futures (such as DI futures in Brazil, if your target country has that) or local-currency bonds (the strike possibly being hard-currency asset swap spreads).
You could hedge some of your unwanted new currency vega with vega to other more liquid assets, e.g. interest rates of another EM currency, that you expect to be correlated. But it's not a good hedge if some idiosyncratic event breaks the correlation, which sometimes happens in EM.
You should further build a collection of stress scenarios (fx pegs breaking, sovereign defaults, foreign invasions, etc), estimate their impact on your book and on the assets I listed above. Yiu could further try to hold some positions- maybe the underlying swap rates, maybe even some equities or key commodities - that, under these stress scenarios, would offset your losses from an unhedged vega. However accountants wouldn't allow you to call this hedge a hedge.
I will relate a couple of tales of rates volatility trades that I've seen first hand, to illustrate why, when you say "insure", you mean market-making, retaining minimum vega, rather than betting on low-probability events like fx pegs breaking, against people who know what's going on better than you.
One large bank took a very large exposure to GRD and EUR interest rates volatilities convergung, shortly before Greece joined the Euro - "insuring" its clients. You may recall that it was unclear until almost the new years whether Greece, then considered EM, would join. If not for some quasi-criminal shenanigans by the then Greek government and another large bank, Greece would not have joined when it did, the rates implied volatilities would have diverged, the clients would get paid a lot, and the bank would have needed yet another taxpayer bailout.
Another time, the same bank got itself very large exposures to MXN rate vol (as well as MXN fx vol, and fx rate, and other MX) the day before U.S. elections. The elections did not go the way the bank bet, the bank lost lots of money, claimed the trades were not authorized, walked some people off the trading floor the next day.
My point is, I don't think you can get paid enough for holding these kinds of risks.
At a larger firm, you may also need to get involved with setting up the XVA, Value at Risk, FRTB-related, etc calculations for your new options. In this case, your firm probably already has procedures driven by regulatory requirements, for dealing conservatively with market factors that have no observable history, such as your new rates implied vols.