Maybe a European Airline like AY (OneWorld) LH (Star Alliance) or KL (Skyteam) as they're more likely to 'common rate' more European destinations with HEL/FRA/AMS for example which are the respective hub cities of the above airlines.
I think it would be cheaper to find an airfare that will get you all the way to Tbilisi rather than London only and electing to pay a sector fare LHR/TSB/LHR. I would ask the tour operator for a ground only price ie the cost of the tour component only ex TBS.
One site I often look at is
FlightStats - Global Flight Tracker, Status Tracking and Airport Information then enter the city & view the departures & arrivals tab to see what airlines fly into/out of that city & from where plus what aircraft type.
For example looking at arrivals into Tbililsi Airport between 0000-1200 shows the following flights. You can only look a few days ahead or in the past so while it won't show you exactly what flights will be operating next year it will give you a good idea of the airlines who fly there.
If I was the traveller I would select TK386 ex IST on a 738, LH2556 ex MUC on a 319 or AZ550 ex FCO on a 320. LO623 ex WAW is maybe a second choice due to an older 734 plane. Some of the other carriers I would travel on in a pink fit.
[TABLE="class: tableListingTable"]
[TR="class: tableHeader"]
[TH="class: tableListingHeaderNoBottomBorder, bgcolor: #EDECEA"][/TH]
[TH="class: tableListingHeaderNoBottomBorder, bgcolor: #EDECEA"][/TH]
[TH="class: tableListingHeaderNoBottomBorder, bgcolor: #EDECEA"]On-time[/TH]
[TH="class: tableListingHeaderNoBottomBorder, bgcolor: #EDECEA"][/TH]
[TH="class: tableListingHeaderNoBottomBorder, bgcolor: #EDECEA, colspan: 2"]Arrival[/TH]
[TH="class: tableListingHeaderNoBottomBorder, bgcolor: #EDECEA"]Term[/TH]
[TH="class: tableListingHeaderNoBottomBorder, bgcolor: #EDECEA"][/TH]
[TH="class: tableListingHeaderNoBottomBorder, bgcolor: #EDECEA"][/TH]
[TH="class: tableListingHeaderNoBottomBorderRightBorder, bgcolor: #EDECEA"][/TH]
[/TR]
[TR="class: tableHeader"]
[TH="class: tableListingHeaderNoTopBorder, bgcolor: #EDECEA"]Origin[/TH]
[TH="class: tableListingHeaderNoTopBorder, bgcolor: #EDECEA"]Flight[/TH]
[TH="class: tableListingHeaderNoTopBorder, bgcolor: #EDECEA"]Rating[/TH]
[TH="class: tableListingHeaderNoTopBorder, bgcolor: #EDECEA"]Airline[/TH]
[TH="class: tableListingHeaderNoTopNoRightBorder, bgcolor: #EDECEA"]Sched[/TH]
[TH="class: tableListingHeaderBottomBorder, bgcolor: #EDECEA"]Actual[/TH]
[TH="class: tableListingHeaderNoTopBorder, bgcolor: #EDECEA"]Gate[/TH]
[TH="class: tableListingHeaderNoTopBorder, bgcolor: #EDECEA"]Status[/TH]
[TH="class: tableListingHeaderNoTopBorder, bgcolor: #EDECEA"]Equip[/TH]
[TH="class: tableListingHeaderNoTopBorderRightBorder, bgcolor: #EDECEA"]Track[/TH]
[/TR]
[TR]
[TD]
SVX Yekaterinburg[/TD]
[TD]
U6 2911[/TD]
[TD][/TD]
[TD]Ural Airlines[/TD]
[TD]12:05 AM[/TD]
[TD="class: tableListingTableNoRightBorder"]1:10 AM ~[/TD]
[TD][/TD]
[TD]En Route
65 min[/TD]
[TD][/TD]
[TD="class: rightCol"]

[/TD]
[/TR]
[TR]
[TD]
SAW Istanbul[/TD]
[TD]
TK 380[/TD]
[TD][/TD]
[TD]Turkish Airlines[/TD]
[TD]12:25 AM[/TD]
[TD="class: tableListingTableNoRightBorder"][/TD]
[TD][/TD]
[TD]Scheduled[/TD]
[TD]738[/TD]
[TD="class: rightCol"]

[/TD]
[/TR]
[TR]
[TD]
TLV Tel Aviv-Yafo[/TD]
[TD]
A9 696[/TD]
[TD][/TD]
[TD]Georgian Airways[/TD]
[TD]12:50 AM[/TD]
[TD="class: tableListingTableNoRightBorder"][/TD]
[TD][/TD]
[TD]En Route[/TD]
[TD]737[/TD]
[TD="class: rightCol"]

[/TD]
[/TR]
[TR]
[TD]
DOK Donetsk[/TD]
[TD]
UJX 510[/TD]
[TD][/TD]
[TD]Atlasjet Ukraine[/TD]
[TD]2:35 AM[/TD]
[TD="class: tableListingTableNoRightBorder"][/TD]
[TD][/TD]
[TD]Scheduled[/TD]
[TD]320[/TD]
[TD="class: rightCol"]

[/TD]
[/TR]
[TR]
[TD]
MSQ Minsk[/TD]
[TD]
B2 735[/TD]
[TD]
[/TD]
[TD]Belavia[/TD]
[TD]2:50 AM[/TD]
[TD="class: tableListingTableNoRightBorder"]2:38 AM[/TD]
[TD][/TD]
[TD]Landed
On-time[/TD]
[TD]E95[/TD]
[TD="class: rightCol"][/TD]
[/TR]
[TR]
[TD]
MUC Munich[/TD]
[TD]
LH 2556[/TD]
[TD]
[/TD]
[TD]Lufthansa[/TD]
[TD]3:00 AM[/TD]
[TD="class: tableListingTableNoRightBorder"]2:56 AM[/TD]
[TD]103[/TD]
[TD]Landed
On-time[/TD]
[TD]319[/TD]
[TD="class: rightCol"][/TD]
[/TR]
[TR]
[TD]
IST Istanbul[/TD]
[TD]
TK 386[/TD]
[TD]
[/TD]
[TD]Turkish Airlines[/TD]
[TD]3:20 AM[/TD]
[TD="class: tableListingTableNoRightBorder"]3:27 AM[/TD]
[TD][/TD]
[TD]Landed
On-time[/TD]
[TD]320[/TD]
[TD="class: rightCol"][/TD]
[/TR]
[TR]
[TD]
WAW Warsaw[/TD]
[TD]
LO 723[/TD]
[TD]
[/TD]
[TD]LOT - Polish Airlines[/TD]
[TD]4:00 AM[/TD]
[TD="class: tableListingTableNoRightBorder"]4:29 AM[/TD]
[TD][/TD]
[TD]Landed
29 min[/TD]
[TD]734[/TD]
[TD="class: rightCol"][/TD]
[/TR]
[TR]
[TD]
FCO Rome[/TD]
[TD]
AZ 550[/TD]
[TD]
[/TD]
[TD]Alitalia[/TD]
[TD]4:00 AM[/TD]
[TD="class: tableListingTableNoRightBorder"]4:10 AM[/TD]
[TD][/TD]
[TD]Landed
On-time[/TD]
[TD]320[/TD]
[TD="class: rightCol"][/TD]
[/TR]
[TR="class: lastRow"]
[TD]
IST Istanbul[/TD]
[TD]
TK 384[/TD]
[TD]
[/TD]
[TD]Turkish Airlines[/TD]
[TD]10:20 AM[/TD]
[TD="class: tableListingTableNoRightBorder"]10:32 AM ~[/TD]
[TD][/TD]
[TD]En Route
On-time[/TD]
[TD]319[/TD]
[TD="class: rightCol"]

[/TD]
[/TR]
[/TABLE]