Is it possible to have an afterburner in a turbofan with a bypass ratio greater than 1:1? At 1.05:1, say, can it have an afterburner?
fooot has pointed out the GE F101 (powers the B-1 Lancer) has a BPR (bypass ratio) of >2:1 and comes with an afterburner.
Other examples are the Turbo-Union RB199 (1.1:1) and the Volvo RM8 (0.97:1). Generally, anything below 4:1 is a low BPR, and bypass air is needed for cooling. Pure turbojets resort to other means of cooling, such as the ram air inlet on a J75, or they run inefficiently at low combustion temperatures.
I can't speak for the inner workings of a classified military engine, but an ideal supersonic engine would be a variable cycle engine (VCE). Operating with bypass in the subsonic regime, and without in the supersonic high-power demand regime.
Such theoretical (to the public at least) examples from the 90's are shown (and linked) below. Do note the idea behind the VCE is the elimination of the wasteful and noisy afterburner.
So, it's an either/or situation. A dual operation of afterburner and high bypass in a non-VCE would not have benefits.
(Flight—PDF) Image shows how a VCE routes the bypass air in both modes of operation. Other variations by RR and Snecma can be seen here. Note the novel design of the Snecma with the free-turbine—extra-fan. VCE's would definitely be heavier and more complex engines.
Can you add reheat to a high bypass turbofan engine? Sure. The real question is what is the purpose for doing so?
The bypass ratio of a turbofan is based largely on its operation regime. Turbojets or low bypass turbofans are excellent choices for operations above Mach 1 to around Mach 2-3. Larger bypass ratios are more suitable to high subsonic and transonic operations. Bypass ratios of 3:1 to 5:1 work quite well for transonic operations, for instance the Rolls Royce BR725 engine with a 4:1 bypass ratio, powering the Bombarider Global Express or Gulfstream G650 at cruise speeds of Mach 0.9. Bypass ratios around 7:1 to 10:1 are great for high subsonic operations around Mach 0.85, which is why they are commonly selected for commercial or transport aircraft which cruise at those speeds.
Reheat offers a lot of thrust, around a 50% increase over max dry thrust, but uses up a lot of fuel, making it unattractive to commercial operators. Military aircraft are willing to sacrifice fuel efficiency for superior thrust in energy intensive operations like ACM, or takeoff with large payloads. As military fighters require supersonic flight for either cruise or dash speeds, a low bypass engine should be utilized for this purpose. Also low bypass ratio engines usually offer a higher engine thrust to weight ratio, in some cases as high as 10:1 versus around 4:1-6:1 for high bypass turbofans, making them very attractive for military applications.
As a result, while it is technically feasible to build a high bypass turbofan with reheat, there is no practical application for it.
In the late 60s/early 70s, P&W proposed an engine (the JTF17) for the projected USA Supersonic Transport (SST) which would have introduced afterburning in the fan-generated section of the airflow: a fan-burner.
This doesn't quite match the sense of the questioner's description but you should always beware of assuming that the most trodden paths are the only ones!