As far as we know, in Ukraine with international seat/bed reservation you simply can't get domestic ticket, it needs to be also international.
In general it became wrong statement.
Such approach really was in the past, but during last years in Ukraine it's allowed to combine domestic ticket with internatinal reservation for travelling domestic leg using international ticket. Travel in domestic train combining with international reservation for domestic leg and domestic ticket is allowed as well.
But seat reservation and ticket are still "mutual serfs of each other" or in other words "non-separable". For example you cannot buy reservation unless you produce original ticket to cashier and cashier will write on ticket and put stamp that reservation to section from A to B sold (so you cannot use same ticket for fishing few reservations coming to different cashiers with same ticket for example), also reservation will contain note that is should be used with ticket number (ticket number). The same story with international reservation for domestic train - they will not sell you it it unless you produce international reservation and only after this they issue the ticket.
I have a sleeper reservation from Lviv to Cierna nad Tisou into that direct sleeper coach Kiyv-Bratislava and I am wondering if I can buy the international ticket only for Chop-Cierna nad Tisou (2 Eur) section and a domestic Ukrainian ticket for the domestic part Lviv-Chop (approx. 100 UAH), or I need to have the international ticket for the whole journey - this is for 12 Eur, so I was thinking I could save something.
No.
It's not allowed to combine domestic ticket with international reservation which is issued for international leg.
And I agree with tabo, depite domestic ticket (which you can buy online) contans both ticket and reservation, but ticket from it is not combinable with international reservation issued for international leg.
If you really on tight budget and seriously need spend less money for this travel - you should better refund your reservation Lviv - Bratislava back if it's possible, and travel using domestic ticket to Chop and travel in seating cars from Chop to Kosice and Bratislava. Sleeping car Kyiv - Bratislava between Chop and Bratislava travel with trains which contains seating cars, so it's possible to cross border here without purchasing international sleeping bed reservation. Also in this case it's worth to think about re-planning your travel using RegioJet train between Kosice and Bratislava if 9 EUR ticket is available on your date.
Or if you prefer having possibility not to exit from train in 6 in the morning in Chop but continue laying (not sure about sleeping, re-gauging workers and custom/border officers anyway will wake you up) - you must cover leg from Lviv to 1st Ukrainian station by international ticket.
The idea of "pendulum" to Uzhhorod proposed by tabo also will work (and probably you have less chances to be waken in Chop in domestic car Kyiv - Uzhhorod than in internatonal car Kyiv - Bratislava) but only the only your preference is not no be waken (but not come to Bratislava faster).