1. Firstly, don't worry about targets. No one can live within their heart rate with a disease like ME, as it involves autonomic dysfunction. (Your autonomic nervous system dictates your pulse rate, as does a plethora of other variables not within your control: Infection, blood volume etc). Living within a pulse rate is only possibly in people with burn our or Chronic Fatigue, AKA Nancy Klimas's ideas she has about getting people to stare at their pulse rate to prevent 'crashes' (doesn't work in ME as you crash sitting still on the sofa, irrespective of movement and pulse rate). This is why I say, don't stress about your heart rate in terms of keeping to a 'goal'. It won't work in a chronic neuroimmune disease like ME, for the reasons given above.
To make you laugh, I have severe ME and without beta blockers my pulse rate is 130 bpm standing up, rising to 180bpm or so after a few mins. Sadly, this is what POTS does to people with ME, and POTS is common in ME and CFS. Another feature of ME and CFS, is that your maximum heart rate is too slow, when exercising. If you are at the higher end of functioning it might be worth considering having someone evaluate your heart rates when walking, exercising gently, or even better have a VO2 max assessment in a specialist clinic or Hospital to see how your body responds to exercise, in a safe environment. (Although rare, some PWME die from exercise and Dr Lerner (CFS specialist in America now deceased) always warned CFS patients never to exercise unless recovered.
2. Every heart rate is safe if it's within normal limits and appropriate for task in hand. Feeling exhausted in ME CFS is not related to pulse rate. Naturally if you try and move and are more exhausted, your pulse rate will usually rise, sometimes dramatically, so an elevated pulse rate (compared to normal) may be a sign how 'sick' you feel on a given day. Again in ME CFS, there is nothing you can do about this and it's part of the pathogenesis of the condition.
3. Your heart might be healthy already, although it may lack blood flow or mitochondrial energy due to the disease process no one understands yet. ME involves mitochondrial dysfunction of a type that is new to Science and is being investigated by at least one group as you read this. Other than chronic infection (possible) this lack of ATP affects cardiac tissue and is also associate to oxidative injury. There is no current information that ME CFS have unhealthy hearts in terms of organ health, other than the fact that oxidative stress (long term) is damaging for the whole body and having a chronically activated immune system will be pouring in cytokines and other chemicals, that long term won't do cardiac tissue any good. Other than takes masses of supplements, there is very little anyone can do to reduce this problem,because again, Science isn't clever enough yet to find out why this can happen in ME CFS.
No one can exercise to get fully fit (build real strength), in ME CFS, if you're being told you can, you're misinformed.
If they could when in a better phase of health and they didn't relapse ever (e.g no severe pay-back in days, weeks, months, years), then they would be recovering anyway (possible but rare) or misdiagnosed.
This doesn't mean you shouldn't exercise, if you're recovering from CFS/Burnout and have no symptoms. The danger comes though, is from relapse. Many patients report they feel 'cured only to relapse once more, sometimes to a worse state than before. It doesn't mean it will happen to you, but it might do. We don't know, and that's why we have to be careful.
Always seek an alternative diagnosis if you feel trapped by your symptoms/body, but it doesn't fit in with the core traits of the diagnosis you have. For example, in CFS the diagnosis is you have 50% or more loss of functioning and that minimal exertion physical or mental leads to relapse for hours, days, weeks, months etc. If you have this, I would not be concerned about heart rates and sticking to a goal, unless you have mild CFS and aren't slammed by your symptoms every day like most (but not all) patients are.
Hence I can't really comment about building your heart up, as that wouldn't be possible in ME CFS as the more severely affected are bed ridden or housebound and others can do gentle walking in bursts and real exercise (literally to build up muscle in a gym), simply isn't possible.
I'd just listen to your body and trust your instincts and take things very slowly. No doctor can promise you that looking t pulse rates can enable higher levels of functioning as there is no Science to support this. Currently there is no diagnostic test for CFS or ME, so literally, no researcher knows who has what.
This is a huge problem when suggestion treatment protocols to patients based on increasing activity, immune therapy or practically anything else.
Hope that helps.