Consider that a conflict like you describe would not be a "total war" with the full resources of the country poured into an effort to win. You cannot sustain something like that for 700 years.
Your war reminds me of the war in Orwell's 1984
the purpose of the unwinnable, perpetual war is to consume human
labour and commodities so that the economy of a superstate cannot
support economic equality, with a high standard of life for every
citizen. By using up most of the produced objects like boots and
rations, the proles are kept poor and uneducated and will neither
realise what the government is doing nor rebel.
That is one spin on a forever war: the grim 1984 spin. The other reason for a war like this is more positive. Nothing unites like a common enemy, and bonds made between soldiers during war last a lifetime. A low level, simmering war of this sort could be used to instill feelings of unity and brotherhood - possibly among different ethnic or religious groups - and so sidestep or smooth over internal issues which otherwise might cause trouble in the governance of the country.
As regards morale for the soldiers, I could imagine that each might do a short tour of duty at the front in doing obligatory military service. The country would not want to make this a hellish experience or one likely to kill a lot of young men. That would be unsustainable. I found this good text describing how the Canadian government took care of its soldiers who went to the far away front in WW1 (and probably a more dangerous endeavor there than the forever war you describe).
Many factors helped persuade soldiers to fight. The bonds of
friendship, loyalty and community based on shared experience and
common dangers were principal among them, leading to a strong sense of
group identity, especially among small groups. Training, firm
discipline and strong leadership also motivated soldiers and, the
threat of punishment helped keep soldiers in line. Military traditions
and values were also important, even in citizen-armies like Canada’s,
and the firm sense that most soldiers shared in the justness of their
cause. The military understood many of the challenges to morale and
the discomfort caused by life in the trenches. It tried to provide
soldiers with the comforts necessary to sustain morale.
You country would do the same. The war is the glue that holds society together. The men who go should be proud to be going, and should return with positive feelings about themselves, their comrades, and the country.