Didn't they have rail supply in Eastern Karelia? why would it be hard to supply them within 50km of the rail line? Of course with Leningrad fallen the Germans would be maximum pressure on the Finns to cut the Murmansk RR at a minimum even if there wasn't an attack on the city itself, probably leveraging food shipments if needed, because with Leningrad down there is no excuse now to.
Now domestically within Finland there is the issue of the civilian government seeing Greater Finland being an option, because with Leningrad down, they really only have Eastern Karelia as an ongoing front and cutting the Murmansk RR would effectively shut down their part of the war other than the outstanding situation around Murmansk itself.
They had rail supply in the southern part of Eastern Karelia, up to a bit north of Karhumäki (Medvedzhegorsk) at the northern end of Lake Onega. North of there, no rail lines outside the prewar Finnish area. So, apart of the Murmansk railway, most of the area of Viena Karelia would be almost entirely dependent on road transport or aircraft. Outside the northern tundra and the Alpine area, this might be some of the most difficult terrain the Axis troops had to tackle in continental Europe, as outside a couple of dry summer months, the few roads here were either made of mud or blocked with snow.
A typical East Karelian road in April 1942:
A better sort of road in August 1941:
The Finnish railways were after the Winter War short of rolling stock, over 10% down from peace time as a lot of locomotives and carriages had to be handed over to the Soviets in 1940. Wartime acquisitions could not replenish the stock to adequate levels. What little was captured from the Soviets in 1941 was often in very poor condition. The Soviets had damaged a lot of their lines and infrastructure when they withdrew, and repairing these was a major challenge for the Finnish railway units, as well as maintaining the lines in difficult conditions was anyway.
All through the advance phase in 1941, the railways were experiencing major difficulties. Only during the static phase of the war, since 1942, could they get the situation somewhat under control, and even then the demands of the military in Karelia were hard to satisfy as Finland has predominately just single-track rail lines, and often major bottlenecks were formed, tens of empty carriages would accumulate on certain stations as they could not be sent west due to the eastbound traffic. Moving a carriage of goods from Western Finland to Eastern Karelia by rail could take two weeks - and that is only to the railhead.
Water transport was of course used on the Ladoga and Onega, but even there the shortage of suitable vessels was significant. On the Onega, for example, captured old paddle steamers were used as auxiliary gunboats in the absense of more appropriate ships.
The Finnish military simply had too much men in arms in 41-44 to supply effectively
on the advance, given the conditions in Karelia and the Finnish transport resources and rail network (which had to supply German units in the north as well). And any more men Finland has in arms, not doing vital civilian work on the home front, the more Finland would be dependent on Germany. So - one of the main objectives of the Finnish government would be to demobilize as much of the military as possible, in case the Soviet threat has diminished. Farms need tending, forests need harvesting and factories need skilled hands. The occupied areas would still need to be occupied, and that alone would mean a lot of men need to stay in uniform anyway. The Finnish logistics system was not capable of supporting a major attack towards the Murmansk railway without crippling (or at least hobbling) the nation in some way and making Finland much more prone to German pressure.
To me this means that the government would commit as little active troops to an attack towards Murmansk as it would think it can get away with. This would not be due to an "excuse" not to attack, but very real logistical, economic and political reasons. Finland is not going out of its way to suck up to the Nazis in a way that hurts national interests, even if the Nazis are winning - as long as the OTL "inner ring" of politicians and soldiers have any say about it.
My estimate is that the Germans would still need to commit significant resources for the attack against Murmansk, as Finland would not commit enough troops to be the major partner in such an effort. This would be the general picture - we can of course haggle over the realistic troop numbers the Finns might send, also in terms of what promises and threats Berlin is ready to make to goad the Finns into taking a bigger role.
Let us consider using the troops on the Karelian Isthmus in an attack against Murmansk. In terms of logistics, the Isthmus is superior to Viena Karelia. A direct and short rail line to southern Finland. A good road network. A number of usable ports on the coast, not least around Viipuri itself. Supplying a force made of even one of the Finnish corps on the Isthmus in 1941, say 2-3 infantry divisions as the main force (plus artillery and engineers, possibly some armor) and a couple of regiments (etc) as a reserve would be much more difficult in Eastern Karelia. All the supply formations would have to be supplied as well, and what ever air assets would be sent to the area. Of course, this does not even take into account what ever troops the Germans would send, or the Finnish troops already in the area. All this at the end of a single rail line, and then just wilderness and poor roads.
Then there is of course the point to be made that the troops that fought initially north of the Ladoga were more suitable for battles in the wilderness than the troops sent to the Isthmus - it is a real concern whether such troops can be found from the three Finnish corps on the Isthmus that could be used in those conditions without risking worse losses than would be acceptable.
There would be the question of morale as well, as many of these men came from the towns in southern Finland and many were Social Democrats or even more left, for example. Meny had already protested crossing the 1920 border in 1941. Now sending them to conquer a frozen Soviet port behind hundreds of kilometers of roadless wilderness could well cause serious morale problems, if not actual mutinies in several units.
As far as I see it, much of the troops that attacked towards Eastern Karelia initially were made of either men from such parts of the country that had roughly similar terrain (Savo and North Karelia) or from areas that were "ideologically trustworthy" (the Ostrobothnian coast and countryside). This is not entirely accidental.