Can you not take the LH flight that leaves MAN at 09:20 direct to FRA? This would give you enough time.

If it was me I wouldn't go for the 08:45 as I'd rather not have to worry about missing a connection. If you were on a single ticket I'd have no problem. However if check-in for LH closes 30 minutes prior to departure then that gives you 1 hour 10. If you are on time that is not a problem as the walk from T3 to T1 is pretty short and immigration doesn't take that long with an EU passport. But if there is a delay then you would be relying on LH to rebook you which they wouldn't be obliged to do and you could end up paying for another ticket (maybe LH aren't as harsh as BD)?