Here's a tip on learning English grammar: English doesn't work by rules. English has a small number of modal verbs and grammatical inflections, with which speakers struggle to communicate a large number of grammatical distinctions. People do their best to suggest whatever temporal distinctions are relevant in a specific situation. The rules in grammar books are just approximations for the many ways in which people try to milk too-few grammatical forms for too-many grammatical distinctions. If you remove the word "backshifting" from your vocabulary, you might find English grammar a little easier to sort out, because you won't be tempted to search for a rule that isn't there.
The way you have to improvise with English tenses is best learned by many examples, preferably in some context that makes them meaningful. Here is one of your counterexamples analyzed, with a little context added:
"It would seem there has been a mistake," said Rupert, looking at the ten-foot-high pile of gravel that had been dumped on the lawn.
Rupert uses the present perfect has been is to indicate that the mistake is relevant now (when Rupert is speaking). The mistake—pouring gravel on the lawn—was made in the past, but the problem of cleaning it up is the salient fact right now. This is a typical case for using the present perfect.
The word would is not being used here to indicate the past tense. Technically, would is the past tense of will, but here it indicates the consequence of a hypothetical condition—in the present. You see how would has more than one grammatical meaning? There is no hypothetical condition stated explicitly. Implicitly, the hypothetical condition is "if the obvious inference that someone dumped a pile of gravel here by mistake is true." In this example, would is chosen for understatement. In many situations where would lacks a condition and describes the present, it's chosen for politeness, since the conditional mood "softens" a statement.
Now here is the original example:
Circus manager: “All you have to do is look the lion in the eye and show him you're not afraid.”
Assistant: “Yes, but the lion would know I was just being deceitful.”
In this example, too, would indicates the consequence of a condition: "If I looked the lion fearlessly in the eye, it would know I was just being deceitful." The assistant is describing a hypothetical future situation.
The reason for choosing was is because "I (am) just being deceitful" is meant in the subjunctive mood. The subordinate clause describes a hypothetical consequence, which the assistant wants to avoid treating as a real situation. A common way to make a future subjunctive in English is to put the verb into the past tense. Thus you see another way that English makes do with a shortage of grammatical forms. This use of the past tense for the future subjunctive works because a listener understands that the sentence is about the future.
Notice that I put "If I looked the lion in the eye…" into the past tense. The same principle is at work. The principle isn't "under such-and-such conditions, you must backshift", it's "English lacks a future subjunctive, and I want to be clear that I mean this verb subjunctively, so I'll just co-opt the past tense to indicate the subjunctive mood. People do that frequently enough that you'll recognize the construction, though it's not a rule."
P.S. Here's another answer, which explains at length how speakers bend would and should to serve many grammatical roles—not following rules, just making reasonable choices among the available words in a wide variety of situations. People choose the tense of a subordinate clause to suggest or reinforce the intended meaning, not according to a rule like "Past-tense modal verbs require subordinate verbs to shift tense."