If you want to maintain consistency of length, casting stroke etc, then the answer is NO. If, on the other hand, you chose to learn the “short belly” line setup on the 7wt and the more standard length system / stroke on the 10/11 you may be able to succeed.
It would require a RIO Windcutter Spey with tips.
If the 10/11 would load up with a WC 9/10/11 full setup, that is Belly, Tip 2 and Tip 1 of whatever density fit the situation the total length of the head would equal 56 feet and weigh 650 Grains.
The 7/8 would require that you remove tip number 2. This would result in a head length of about 38 feet and somewhere around 500 Grains. If the 7/8 is in the 12 to 13 foot range this is a very comfortable system to cast, ala Mike Kinney style.
Do you already have the reel, or would you be shopping. How long is the 10/11 you have in mind ?? I too prefer a balance between rod and reel, but a limited budget is a limited budget. Email me if you want to take this offline.
Josh, I agree with Rich in general, but I feel rather more strongly that trying to use the same line for such very different rods will steal much of the joy of spey casting and may slow your progress to becoming a good spey caster. Rich has tried to give you a solution that might be doable, but I have found that spey rods are far more sensitive to line weight variations than single handers. A well matched spey combination is an absolute joy to cast. Add or remove just 5-10 grains and even a reasonably competent caster is in for some frustration. That frustration will be exponentially greater for those who are new to two-handed rods. Sure you can do it, just like you can cast a 2 weight line on a 9 weight single-hander, but...
I recommend that to start, you spend you limited funds on a good line for one of the two rods you named rather than trying to force a single line to work on both rods. Dialed in with the right line the 7/8 could work for most winter and summer fishing.